Theorem ceqsex2v 2772

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. x E. y ( x = A /\ y = B /\ ph ) <-> ch )

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

Allowed substitution hints:   ph( x, y)   ps( y)   ch( x)

Proof of Theorem ceqsex2v

Step	Hyp	Ref	Expression
1		nfv 1522	. 2 |- F/ x ps
2		nfv 1522	. 2 |- F/ y ch
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 2771	1 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 974   = wceq 1349   E.wex 1486   e. wcel 2142   _Vcvv 2731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-3an 976  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-v 2733

This theorem is referenced by:  ceqsex3v  2773  ceqsex4v  2774