Theorem ceqsex2v 2767

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 1516	. 2 |- F/ x ps
2		nfv 1516	. 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 2766	1 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 968   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  ceqsex3v  2768  ceqsex4v  2769