Theorem gencbvex2 2601

Description: Restatement of gencbvex 2600 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)

Hypotheses

Ref	Expression
gencbvex2.1	|- A e. _V
gencbvex2.2	|- ( A = y -> ( ph <-> ps ) )
gencbvex2.3	|- ( A = y -> ( ch <-> th ) )
gencbvex2.4	|- ( th -> E. x ( ch /\ A = y ) )

Assertion

Ref	Expression
gencbvex2	|- ( 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 gencbvex2

Step	Hyp	Ref	Expression
1		gencbvex2.1	. 2 |- A e. _V
2		gencbvex2.2	. 2 |- ( A = y -> ( ph <-> ps ) )
3		gencbvex2.3	. 2 |- ( A = y -> ( ch <-> th ) )
4		gencbvex2.4	. . 3 |- ( th -> E. x ( ch /\ A = y ) )
5	3	biimpac 282	. . . 4 |- ( ( ch /\ A = y ) -> th )
6	5	exlimiv 1489	. . 3 |- ( E. x ( ch /\ A = y ) -> th )
7	4, 6	impbii 117	. 2 |- ( th <-> E. x ( ch /\ A = y ) )
8	1, 2, 3, 7	gencbvex 2600	1 |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by: (None)