Theorem ceqsex2v 3245

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 1843	. 2 |- F/ x ps
2		nfv 1843	. 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 3244	1 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  ceqsex3v  3246  ceqsex4v  3247  ispos  16947  elfuns  32022  brimg  32044  brapply  32045  brsuccf  32048  brrestrict  32056  dfrdg4  32058  diblsmopel  36460