Theorem ceqsex2 2800

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 984	. . . . 5 |- ( ( x = A /\ y = B /\ ph ) <-> ( x = A /\ ( y = B /\ ph ) ) )
2	1	exbii 1616	. . . 4 |- ( E. y ( x = A /\ y = B /\ ph ) <-> E. y ( x = A /\ ( y = B /\ ph ) ) )
3		19.42v 1918	. . . 4 |- ( E. y ( x = A /\ ( y = B /\ ph ) ) <-> ( x = A /\ E. y ( y = B /\ ph ) ) )
4	2, 3	bitri 184	. . 3 |- ( E. y ( x = A /\ y = B /\ ph ) <-> ( x = A /\ E. y ( y = B /\ ph ) ) )
5	4	exbii 1616	. 2 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
6		nfv 1539	. . . . 5 |- F/ x y = B
7		ceqsex2.1	. . . . 5 |- F/ x ps
8	6, 7	nfan 1576	. . . 4 |- F/ x ( y = B /\ ps )
9	8	nfex 1648	. . 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 464	. . . 4 |- ( x = A -> ( ( y = B /\ ph ) <-> ( y = B /\ ps ) ) )
13	12	exbidv 1836	. . 3 |- ( x = A -> ( E. y ( y = B /\ ph ) <-> E. y ( y = B /\ ps ) ) )
14	9, 10, 13	ceqsex 2798	. 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 2798	. 2 |- ( E. y ( y = B /\ ps ) <-> ch )
19	5, 14, 18	3bitri 206	1 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   F/wnf 1471   E.wex 1503   e. wcel 2164   _Vcvv 2760

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-v 2762

This theorem is referenced by:  ceqsex2v  2801