Theorem gencbvex2 2902

Description: Restatement of gencbvex 2901 with weaker hypotheses. (Contributed by Jeffrey 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 472	. . . 4 |- ((ch /\ A = y) -> th)
6	5	exlimiv 1634	. . 3 |- (E.x(ch /\ A = y) -> th)
7	4, 6	impbii 180	. 2 |- (th <-> E.x(ch /\ A = y))
8	1, 2, 3, 7	gencbvex 2901	1 |- (E.x(ch /\ ph) <-> E.y(th /\ ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by: (None)