Theorem gencbvex 2645

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

Hypotheses

Ref	Expression
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

Ref	Expression
gencbvex	|- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )

Distinct variable groups:   ps, x   ph, y   th, x   ch, y   y, A

Allowed substitution hints:   ph( x)   ps( y)   ch( x)   th( y)   A( x)

Proof of Theorem gencbvex

Step	Hyp	Ref	Expression
1		excom 1594	. 2 |- ( E. x E. y ( y = A /\ ( th /\ ps ) ) <-> E. y E. x ( y = A /\ ( th /\ ps ) ) )
2		gencbvex.1	. . . 4 |- A e. _V
3		gencbvex.3	. . . . . . 7 |- ( A = y -> ( ch <-> th ) )
4		gencbvex.2	. . . . . . 7 |- ( A = y -> ( ph <-> ps ) )
5	3, 4	anbi12d 456	. . . . . 6 |- ( A = y -> ( ( ch /\ ph ) <-> ( th /\ ps ) ) )
6	5	bicomd 139	. . . . 5 |- ( A = y -> ( ( th /\ ps ) <-> ( ch /\ ph ) ) )
7	6	eqcoms 2084	. . . 4 |- ( y = A -> ( ( th /\ ps ) <-> ( ch /\ ph ) ) )
8	2, 7	ceqsexv 2638	. . 3 |- ( E. y ( y = A /\ ( th /\ ps ) ) <-> ( ch /\ ph ) )
9	8	exbii 1536	. 2 |- ( E. x E. y ( y = A /\ ( th /\ ps ) ) <-> E. x ( ch /\ ph ) )
10		19.41v 1823	. . . 4 |- ( E. x ( y = A /\ ( th /\ ps ) ) <-> ( E. x y = A /\ ( th /\ ps ) ) )
11		simpr 108	. . . . 5 |- ( ( E. x y = A /\ ( th /\ ps ) ) -> ( th /\ ps ) )
12		gencbvex.4	. . . . . . . 8 |- ( th <-> E. x ( ch /\ A = y ) )
13		eqcom 2083	. . . . . . . . . . 11 |- ( A = y <-> y = A )
14	13	biimpi 118	. . . . . . . . . 10 |- ( A = y -> y = A )
15	14	adantl 271	. . . . . . . . 9 |- ( ( ch /\ A = y ) -> y = A )
16	15	eximi 1531	. . . . . . . 8 |- ( E. x ( ch /\ A = y ) -> E. x y = A )
17	12, 16	sylbi 119	. . . . . . 7 |- ( th -> E. x y = A )
18	17	adantr 270	. . . . . 6 |- ( ( th /\ ps ) -> E. x y = A )
19	18	ancri 317	. . . . 5 |- ( ( th /\ ps ) -> ( E. x y = A /\ ( th /\ ps ) ) )
20	11, 19	impbii 124	. . . 4 |- ( ( E. x y = A /\ ( th /\ ps ) ) <-> ( th /\ ps ) )
21	10, 20	bitri 182	. . 3 |- ( E. x ( y = A /\ ( th /\ ps ) ) <-> ( th /\ ps ) )
22	21	exbii 1536	. 2 |- ( E. y E. x ( y = A /\ ( th /\ ps ) ) <-> E. y ( th /\ ps ) )
23	1, 9, 22	3bitr3i 208	1 |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  gencbvex2  2646