Theorem ceqsex2 2639

Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Hypotheses

Ref	Expression
ceqsex2.1	|- F/ x ps
ceqsex2.2	|- F/ y ch
ceqsex2.3	|- A e. _V
ceqsex2.4	|- B e. _V
ceqsex2.5	|- ( x = A -> ( ph <-> ps ) )
ceqsex2.6	|- ( y = B -> ( ps <-> ch ) )

Assertion

Ref	Expression
ceqsex2	|- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )

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

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

Proof of Theorem ceqsex2

Step	Hyp	Ref	Expression
1		3anass 923	. . . . 5 |- ( ( x = A /\ y = B /\ ph ) <-> ( x = A /\ ( y = B /\ ph ) ) )
2	1	exbii 1536	. . . 4 |- ( E. y ( x = A /\ y = B /\ ph ) <-> E. y ( x = A /\ ( y = B /\ ph ) ) )
3		19.42v 1827	. . . 4 |- ( E. y ( x = A /\ ( y = B /\ ph ) ) <-> ( x = A /\ E. y ( y = B /\ ph ) ) )
4	2, 3	bitri 182	. . 3 |- ( E. y ( x = A /\ y = B /\ ph ) <-> ( x = A /\ E. y ( y = B /\ ph ) ) )
5	4	exbii 1536	. 2 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
6		nfv 1461	. . . . 5 |- F/ x y = B
7		ceqsex2.1	. . . . 5 |- F/ x ps
8	6, 7	nfan 1497	. . . 4 |- F/ x ( y = B /\ ps )
9	8	nfex 1568	. . 3 |- F/ x E. y ( y = B /\ ps )
10		ceqsex2.3	. . 3 |- A e. _V
11		ceqsex2.5	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
12	11	anbi2d 451	. . . 4 |- ( x = A -> ( ( y = B /\ ph ) <-> ( y = B /\ ps ) ) )
13	12	exbidv 1746	. . 3 |- ( x = A -> ( E. y ( y = B /\ ph ) <-> E. y ( y = B /\ ps ) ) )
14	9, 10, 13	ceqsex 2637	. 2 |- ( E. x ( x = A /\ E. y ( y = B /\ ph ) ) <-> E. y ( y = B /\ ps ) )
15		ceqsex2.2	. . 3 |- F/ y ch
16		ceqsex2.4	. . 3 |- B e. _V
17		ceqsex2.6	. . 3 |- ( y = B -> ( ps <-> ch ) )
18	15, 16, 17	ceqsex 2637	. 2 |- ( E. y ( y = B /\ ps ) <-> ch )
19	5, 14, 18	3bitri 204	1 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   F/wnf 1389   E.wex 1421   e. wcel 1433   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  ceqsex2v  2640