Theorem gencbvex2 3251

Description: Restatement of gencbvex 3250 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 503	. . . 4 |- ( ( ch /\ A = y ) -> th )
6	5	exlimiv 1858	. . 3 |- ( E. x ( ch /\ A = y ) -> th )
7	4, 6	impbii 199	. 2 |- ( th <-> E. x ( ch /\ A = y ) )
8	1, 2, 3, 7	gencbvex 3250	1 |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)