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Metamath Proof Explorer

Theorem cgsex4gOLD

Description: Obsolete version of cgsex4g as of 21-Mar-2025. (Contributed by NM, 5-Aug-1995) Avoid ax-10 , ax-11 . (Revised by GG, 28-Jun-2024) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses cgsex4gOLD.1 
|- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch )
cgsex4gOLD.2 
|- ( ch -> ( ph <-> ps ) )
Assertion cgsex4gOLD 
|- ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) )

Proof

Step Hyp Ref Expression
1 cgsex4gOLD.1 
 |-  ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch )
2 cgsex4gOLD.2 
 |-  ( ch -> ( ph <-> ps ) )
3 2 biimpa 
 |-  ( ( ch /\ ph ) -> ps )
4 3 exlimivv 
 |-  ( E. z E. w ( ch /\ ph ) -> ps )
5 4 exlimivv 
 |-  ( E. x E. y E. z E. w ( ch /\ ph ) -> ps )
6 elisset 
 |-  ( A e. R -> E. x x = A )
7 elisset 
 |-  ( B e. S -> E. y y = B )
8 6 7 anim12i 
 |-  ( ( A e. R /\ B e. S ) -> ( E. x x = A /\ E. y y = B ) )
9 exdistrv 
 |-  ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
10 8 9 sylibr 
 |-  ( ( A e. R /\ B e. S ) -> E. x E. y ( x = A /\ y = B ) )
11 elisset 
 |-  ( C e. R -> E. z z = C )
12 elisset 
 |-  ( D e. S -> E. w w = D )
13 11 12 anim12i 
 |-  ( ( C e. R /\ D e. S ) -> ( E. z z = C /\ E. w w = D ) )
14 exdistrv 
 |-  ( E. z E. w ( z = C /\ w = D ) <-> ( E. z z = C /\ E. w w = D ) )
15 13 14 sylibr 
 |-  ( ( C e. R /\ D e. S ) -> E. z E. w ( z = C /\ w = D ) )
16 10 15 anim12i 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) )
17 eqeq1 
 |-  ( y = v -> ( y = B <-> v = B ) )
18 17 anbi2d 
 |-  ( y = v -> ( ( x = A /\ y = B ) <-> ( x = A /\ v = B ) ) )
19 18 anbi1d 
 |-  ( y = v -> ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> ( ( x = A /\ v = B ) /\ ( z = C /\ w = D ) ) ) )
20 19 exbidv 
 |-  ( y = v -> ( E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> E. w ( ( x = A /\ v = B ) /\ ( z = C /\ w = D ) ) ) )
21 20 notbid 
 |-  ( y = v -> ( -. E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> -. E. w ( ( x = A /\ v = B ) /\ ( z = C /\ w = D ) ) ) )
22 eqeq1 
 |-  ( z = v -> ( z = C <-> v = C ) )
23 22 anbi1d 
 |-  ( z = v -> ( ( z = C /\ w = D ) <-> ( v = C /\ w = D ) ) )
24 23 anbi2d 
 |-  ( z = v -> ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> ( ( x = A /\ y = B ) /\ ( v = C /\ w = D ) ) ) )
25 24 exbidv 
 |-  ( z = v -> ( E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> E. w ( ( x = A /\ y = B ) /\ ( v = C /\ w = D ) ) ) )
26 25 notbid 
 |-  ( z = v -> ( -. E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> -. E. w ( ( x = A /\ y = B ) /\ ( v = C /\ w = D ) ) ) )
27 21 26 alcomw 
 |-  ( A. y A. z -. E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> A. z A. y -. E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
28 27 notbii 
 |-  ( -. A. y A. z -. E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> -. A. z A. y -. E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
29 2exnaln 
 |-  ( E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> -. A. y A. z -. E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
30 2exnaln 
 |-  ( E. z E. y E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> -. A. z A. y -. E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
31 28 29 30 3bitr4i 
 |-  ( E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> E. z E. y E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
32 31 exbii 
 |-  ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> E. x E. z E. y E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
33 4exdistrv 
 |-  ( E. x E. z E. y E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> ( E. x E. y ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) )
34 32 33 bitri 
 |-  ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> ( E. x E. y ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) )
35 16 34 sylibr 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) )
36 1 2eximi 
 |-  ( E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> E. z E. w ch )
37 36 2eximi 
 |-  ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> E. x E. y E. z E. w ch )
38 35 37 syl 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> E. x E. y E. z E. w ch )
39 2 biimprcd 
 |-  ( ps -> ( ch -> ph ) )
40 39 ancld 
 |-  ( ps -> ( ch -> ( ch /\ ph ) ) )
41 40 2eximdv 
 |-  ( ps -> ( E. z E. w ch -> E. z E. w ( ch /\ ph ) ) )
42 41 2eximdv 
 |-  ( ps -> ( E. x E. y E. z E. w ch -> E. x E. y E. z E. w ( ch /\ ph ) ) )
43 38 42 syl5com 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( ps -> E. x E. y E. z E. w ( ch /\ ph ) ) )
44 5 43 impbid2 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) )