Theorem ceqsex2 2895

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

Hypotheses

Ref	Expression
ceqsex2.1	|- F/xps
ceqsex2.2	|- F/ych
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.xE.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 938	. . . . 5 |- ((x = A /\ y = B /\ ph) <-> (x = A /\ (y = B /\ ph)))
2	1	exbii 1582	. . . 4 |- (E.y(x = A /\ y = B /\ ph) <-> E.y(x = A /\ (y = B /\ ph)))
3		19.42v 1905	. . . 4 |- (E.y(x = A /\ (y = B /\ ph)) <-> (x = A /\ E.y(y = B /\ ph)))
4	2, 3	bitri 240	. . 3 |- (E.y(x = A /\ y = B /\ ph) <-> (x = A /\ E.y(y = B /\ ph)))
5	4	exbii 1582	. 2 |- (E.xE.y(x = A /\ y = B /\ ph) <-> E.x(x = A /\ E.y(y = B /\ ph)))
6		nfv 1619	. . . . 5 |- F/x y = B
7		ceqsex2.1	. . . . 5 |- F/xps
8	6, 7	nfan 1824	. . . 4 |- F/x(y = B /\ ps)
9	8	nfex 1843	. . 3 |- F/xE.y(y = B /\ ps)
10		ceqsex2.3	. . 3 |- A e. _V
11		ceqsex2.5	. . . . 5 |- (x = A -> (ph <-> ps))
12	11	anbi2d 684	. . . 4 |- (x = A -> ((y = B /\ ph) <-> (y = B /\ ps)))
13	12	exbidv 1626	. . 3 |- (x = A -> (E.y(y = B /\ ph) <-> E.y(y = B /\ ps)))
14	9, 10, 13	ceqsex 2893	. 2 |- (E.x(x = A /\ E.y(y = B /\ ph)) <-> E.y(y = B /\ ps))
15		ceqsex2.2	. . 3 |- F/ych
16		ceqsex2.4	. . 3 |- B e. _V
17		ceqsex2.6	. . 3 |- (y = B -> (ps <-> ch))
18	15, 16, 17	ceqsex 2893	. 2 |- (E.y(y = B /\ ps) <-> ch)
19	5, 14, 18	3bitri 262	1 |- (E.xE.y(x = A /\ y = B /\ ph) <-> ch)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710  _Vcvv 2859

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 2861

This theorem is referenced by:  ceqsex2v  2896