Theorem ceqsex2 3072

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 1011	. . . . 5 |- ( ( x = A /\ y = B /\ ph ) <-> ( x = A /\ ( y = B /\ ph ) ) )
2	1	exbii 1726	. . . 4 |- ( E. y ( x = A /\ y = B /\ ph ) <-> E. y ( x = A /\ ( y = B /\ ph ) ) )
3		19.42v 1842	. . . 4 |- ( E. y ( x = A /\ ( y = B /\ ph ) ) <-> ( x = A /\ E. y ( y = B /\ ph ) ) )
4	2, 3	bitri 257	. . 3 |- ( E. y ( x = A /\ y = B /\ ph ) <-> ( x = A /\ E. y ( y = B /\ ph ) ) )
5	4	exbii 1726	. 2 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> E. x ( x = A /\ E. y ( y = B /\ ph ) ) )
6		nfv 1769	. . . . 5 |- F/ x y = B
7		ceqsex2.1	. . . . 5 |- F/ x ps
8	6, 7	nfan 2031	. . . 4 |- F/ x ( y = B /\ ps )
9	8	nfex 2050	. . 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 718	. . . 4 |- ( x = A -> ( ( y = B /\ ph ) <-> ( y = B /\ ps ) ) )
13	12	exbidv 1776	. . 3 |- ( x = A -> ( E. y ( y = B /\ ph ) <-> E. y ( y = B /\ ps ) ) )
14	9, 10, 13	ceqsex 3069	. 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 3069	. 2 |- ( E. y ( y = B /\ ps ) <-> ch )
19	5, 14, 18	3bitri 279	1 |- ( E. x E. y ( x = A /\ y = B /\ ph ) <-> ch )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   F/wnf 1675   e. wcel 1904   _Vcvv 3031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033

This theorem is referenced by:  ceqsex2v  3073