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Metamath Proof Explorer

Theorem gencbvex

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996) (Proof shortened by Andrew Salmon, 8-Jun-2011)

Ref Expression
Hypotheses gencbvex.1 
|- A e. _V
gencbvex.2 
|- ( A = y -> ( ph <-> ps ) )
gencbvex.3 
|- ( A = y -> ( ch <-> th ) )
gencbvex.4 
|- ( th <-> E. x ( ch /\ A = y ) )
Assertion gencbvex 
|- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )

Proof

Step Hyp Ref Expression
1 gencbvex.1 
 |-  A e. _V
2 gencbvex.2 
 |-  ( A = y -> ( ph <-> ps ) )
3 gencbvex.3 
 |-  ( A = y -> ( ch <-> th ) )
4 gencbvex.4 
 |-  ( th <-> E. x ( ch /\ A = y ) )
5 excom 
 |-  ( E. x E. y ( y = A /\ ( th /\ ps ) ) <-> E. y E. x ( y = A /\ ( th /\ ps ) ) )
6 3 2 anbi12d 
 |-  ( A = y -> ( ( ch /\ ph ) <-> ( th /\ ps ) ) )
7 6 bicomd 
 |-  ( A = y -> ( ( th /\ ps ) <-> ( ch /\ ph ) ) )
8 7 eqcoms 
 |-  ( y = A -> ( ( th /\ ps ) <-> ( ch /\ ph ) ) )
9 1 8 ceqsexv 
 |-  ( E. y ( y = A /\ ( th /\ ps ) ) <-> ( ch /\ ph ) )
10 9 exbii 
 |-  ( E. x E. y ( y = A /\ ( th /\ ps ) ) <-> E. x ( ch /\ ph ) )
11 19.41v 
 |-  ( E. x ( y = A /\ ( th /\ ps ) ) <-> ( E. x y = A /\ ( th /\ ps ) ) )
12 simpr 
 |-  ( ( E. x y = A /\ ( th /\ ps ) ) -> ( th /\ ps ) )
13 eqcom 
 |-  ( A = y <-> y = A )
14 13 biimpi 
 |-  ( A = y -> y = A )
15 14 adantl 
 |-  ( ( ch /\ A = y ) -> y = A )
16 15 eximi 
 |-  ( E. x ( ch /\ A = y ) -> E. x y = A )
17 4 16 sylbi 
 |-  ( th -> E. x y = A )
18 17 adantr 
 |-  ( ( th /\ ps ) -> E. x y = A )
19 18 ancri 
 |-  ( ( th /\ ps ) -> ( E. x y = A /\ ( th /\ ps ) ) )
20 12 19 impbii 
 |-  ( ( E. x y = A /\ ( th /\ ps ) ) <-> ( th /\ ps ) )
21 11 20 bitri 
 |-  ( E. x ( y = A /\ ( th /\ ps ) ) <-> ( th /\ ps ) )
22 21 exbii 
 |-  ( E. y E. x ( y = A /\ ( th /\ ps ) ) <-> E. y ( th /\ ps ) )
23 5 10 22 3bitr3i 
 |-  ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )