Theorem ceqsex2v 1837

Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Hypotheses

Ref	Expression
ceqsex2v.1	|- A e. V
ceqsex2v.2	|- B e. V
ceqsex2v.3	|- (x = A -> (ph <-> ps))
ceqsex2v.4	|- (y = B -> (ps <-> ch))

Assertion

Ref	Expression
ceqsex2v	|- (E.xE.y((x = A /\ y = B) /\ ph) <-> ch)

Distinct variable groups:   x,y,A   x,B,y   ch,y   ps,x

Proof of Theorem ceqsex2v

Step	Hyp	Ref	Expression
1		ax-17 971	. 2 |- (ps -> A.xps)
2		ax-17 971	. 2 |- (ch -> A.ych)
3		ceqsex2v.1	. 2 |- A e. V
4		ceqsex2v.2	. 2 |- B e. V
5		ceqsex2v.3	. 2 |- (x = A -> (ph <-> ps))
6		ceqsex2v.4	. 2 |- (y = B -> (ps <-> ch))
7	1, 2, 3, 4, 5, 6	ceqsex2 1836	1 |- (E.xE.y((x = A /\ y = B) /\ ph) <-> ch)

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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