Theorem gencbvex2 2669

Description: Restatement of gencbvex 2668 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 293	. . . 4 |- ( ( ch /\ A = y ) -> th )
6	5	exlimiv 1535	. . 3 |- ( E. x ( ch /\ A = y ) -> th )
7	4, 6	impbii 125	. 2 |- ( th <-> E. x ( ch /\ A = y ) )
8	1, 2, 3, 7	gencbvex 2668	1 |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   E.wex 1427   e. wcel 1439   _Vcvv 2622

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-v 2624

This theorem is referenced by: (None)