Theorem gencbvex2 1839

Description: Restatement of gencbvex 1838 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

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 418	. . . 4 |- ((ch /\ A = y) -> th)
6	5	19.23aiv 1295	. . 3 |- (E.x(ch /\ A = y) -> th)
7	4, 6	impbi 157	. 2 |- (th <-> E.x(ch /\ A = y))
8	1, 2, 3, 7	gencbvex 1838	1 |- (E.x(ch /\ ph) <-> E.y(th /\ ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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