Theorem gencbvex2 3060

Description: Restatement of gencbvex 3059 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 493	. . . 4 |- ( ( ch /\ A = y ) -> th )
6	5	exlimiv 1779	. . 3 |- ( E. x ( ch /\ A = y ) -> th )
7	4, 6	impbii 192	. 2 |- ( th <-> E. x ( ch /\ A = y ) )
8	1, 2, 3, 7	gencbvex 3059	1 |- ( E. x ( ch /\ ph ) <-> E. y ( th /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1447   E.wex 1666   e. wcel 1890   _Vcvv 3012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-ext 2431

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3014

This theorem is referenced by: (None)