Theorem ceqsex2v 2842

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 1574	. 2 |- F/ x ps
2		nfv 1574	. 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 2841	1 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by:  ceqsex3v  2843  ceqsex4v  2844