Theorem ceqsex2v 2897

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   ps,x   ch,y

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

Proof of Theorem ceqsex2v

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 |- F/xps
2		nfv 1619	. 2 |- F/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 2896	1 |- (E.xE.y(x = A /\ y = B /\ ph) <-> ch)

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  ceqsex3v  2898  ceqsex4v  2899  opksnelsik  4266  sikexlem  4296  br1stg  4731  elswap  4741  brswap2  4861  brsnsi  5774  oqelins4  5795  dmpprod  5841  lecex  6116  addccan2nclem1  6264  nmembers1lem1  6269