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Metamath Proof Explorer

Theorem cgsex4g

Description: An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995) Avoid ax-10 , ax-11 . (Revised by GG, 28-Jun-2024) Avoid ax-9 , ax-ext . (Revised by Wolf Lammen, 21-Mar-2025)

Ref Expression
Hypotheses cgsex4g.1 
|- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch )
cgsex4g.2 
|- ( ch -> ( ph <-> ps ) )
Assertion cgsex4g 
|- ( ( ( 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 cgsex4g.1 
 |-  ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> ch )
2 cgsex4g.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 elisset 
 |-  ( C e. R -> E. z z = C )
10 elisset 
 |-  ( D e. S -> E. w w = D )
11 9 10 anim12i 
 |-  ( ( C e. R /\ D e. S ) -> ( E. z z = C /\ E. w w = D ) )
12 8 11 anim12i 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( ( E. x x = A /\ E. y y = B ) /\ ( E. z z = C /\ E. w w = D ) ) )
13 19.42vv 
 |-  ( E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> ( ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) )
14 13 2exbii 
 |-  ( 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 ) ) )
15 19.41vv 
 |-  ( E. x E. y ( ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) <-> ( E. x E. y ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) )
16 exdistrv 
 |-  ( E. x E. y ( x = A /\ y = B ) <-> ( E. x x = A /\ E. y y = B ) )
17 exdistrv 
 |-  ( E. z E. w ( z = C /\ w = D ) <-> ( E. z z = C /\ E. w w = D ) )
18 16 17 anbi12i 
 |-  ( ( E. x E. y ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D ) ) <-> ( ( E. x x = A /\ E. y y = B ) /\ ( E. z z = C /\ E. w w = D ) ) )
19 14 15 18 3bitri 
 |-  ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) <-> ( ( E. x x = A /\ E. y y = B ) /\ ( E. z z = C /\ E. w w = D ) ) )
20 12 19 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 ) ) )
21 1 2eximi 
 |-  ( E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) -> E. z E. w ch )
22 21 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 )
23 20 22 syl 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> E. x E. y E. z E. w ch )
24 2 biimprcd 
 |-  ( ps -> ( ch -> ph ) )
25 24 ancld 
 |-  ( ps -> ( ch -> ( ch /\ ph ) ) )
26 25 2eximdv 
 |-  ( ps -> ( E. z E. w ch -> E. z E. w ( ch /\ ph ) ) )
27 26 2eximdv 
 |-  ( ps -> ( E. x E. y E. z E. w ch -> E. x E. y E. z E. w ( ch /\ ph ) ) )
28 23 27 syl5com 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( ps -> E. x E. y E. z E. w ( ch /\ ph ) ) )
29 5 28 impbid2 
 |-  ( ( ( A e. R /\ B e. S ) /\ ( C e. R /\ D e. S ) ) -> ( E. x E. y E. z E. w ( ch /\ ph ) <-> ps ) )