Theorem ceqsex2 1836

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

Hypotheses

Ref	Expression
ceqsex2.1	|- (ps -> A.xps)
ceqsex2.2	|- (ch -> A.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

Proof of Theorem ceqsex2

Step	Hyp	Ref	Expression
1		anass 439	. . . . 5 |- (((x = A /\ y = B) /\ ph) <-> (x = A /\ (y = B /\ ph)))
2	1	exbii 1051	. . . 4 |- (E.y((x = A /\ y = B) /\ ph) <-> E.y(x = A /\ (y = B /\ ph)))
3		19.42v 1308	. . . 4 |- (E.y(x = A /\ (y = B /\ ph)) <-> (x = A /\ E.y(y = B /\ ph)))
4	2, 3	bitr 173	. . 3 |- (E.y((x = A /\ y = B) /\ ph) <-> (x = A /\ E.y(y = B /\ ph)))
5	4	exbii 1051	. 2 |- (E.xE.y((x = A /\ y = B) /\ ph) <-> E.x(x = A /\ E.y(y = B /\ ph)))
6		ax-17 971	. . . . 5 |- (y = B -> A.x y = B)
7		ceqsex2.1	. . . . 5 |- (ps -> A.xps)
8	6, 7	hban 1009	. . . 4 |- ((y = B /\ ps) -> A.x(y = B /\ ps))
9	8	hbex 1006	. . 3 |- (E.y(y = B /\ ps) -> A.xE.y(y = B /\ ps))
10		ceqsex2.3	. . 3 |- A e. V
11		ceqsex2.5	. . . . 5 |- (x = A -> (ph <-> ps))
12	11	anbi2d 616	. . . 4 |- (x = A -> ((y = B /\ ph) <-> (y = B /\ ps)))
13	12	exbidv 1279	. . 3 |- (x = A -> (E.y(y = B /\ ph) <-> E.y(y = B /\ ps)))
14	9, 10, 13	ceqsex 1834	. 2 |- (E.x(x = A /\ E.y(y = B /\ ph)) <-> E.y(y = B /\ ps))
15		ceqsex2.2	. . 3 |- (ch -> A.ych)
16		ceqsex2.4	. . 3 |- B e. V
17		ceqsex2.6	. . 3 |- (y = B -> (ps <-> ch))
18	15, 16, 17	ceqsex 1834	. 2 |- (E.y(y = B /\ ps) <-> ch)
19	5, 14, 18	3bitr 177	1 |- (E.xE.y((x = A /\ y = B) /\ ph) <-> ch)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem is referenced by:  ceqsex2v 1837

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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