Theorem copsex2g 2793

Description: Implicit substitution inference for ordered pairs.

Hypothesis

Ref	Expression
copsex2g.1	|- ((x = A /\ y = B) -> (ph <-> ps))

Assertion

Ref	Expression
copsex2g	|- ((A e. C /\ B e. D) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))

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

Proof of Theorem copsex2g

Step	Hyp	Ref	Expression
1		eeanv 1323	. . 3 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
2		hbe1 1016	. . . . 5 |- (E.xE.y(<.A, B>. = <.x, y>. /\ ph) -> A.xE.xE.y(<.A, B>. = <.x, y>. /\ ph))
3		ax-17 971	. . . . 5 |- (ps -> A.xps)
4	2, 3	hbbi 1010	. . . 4 |- ((E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps) -> A.x(E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
5		hbe1 1016	. . . . . . 7 |- (E.y(<.A, B>. = <.x, y>. /\ ph) -> A.yE.y(<.A, B>. = <.x, y>. /\ ph))
6	5	hbex 1006	. . . . . 6 |- (E.xE.y(<.A, B>. = <.x, y>. /\ ph) -> A.yE.xE.y(<.A, B>. = <.x, y>. /\ ph))
7		ax-17 971	. . . . . 6 |- (ps -> A.yps)
8	6, 7	hbbi 1010	. . . . 5 |- ((E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps) -> A.y(E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
9		opeq12 2489	. . . . . . 7 |- ((x = A /\ y = B) -> <.x, y>. = <.A, B>.)
10		copsexg 2792	. . . . . . . 8 |- (<.A, B>. = <.x, y>. -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
11	10	eqcoms 1478	. . . . . . 7 |- (<.x, y>. = <.A, B>. -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
12	9, 11	syl 10	. . . . . 6 |- ((x = A /\ y = B) -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
13		copsex2g.1	. . . . . 6 |- ((x = A /\ y = B) -> (ph <-> ps))
14	12, 13	bitr3d 530	. . . . 5 |- ((x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
15	8, 14	19.23ai 1064	. . . 4 |- (E.y(x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
16	4, 15	19.23ai 1064	. . 3 |- (E.xE.y(x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
17	1, 16	sylbir 201	. 2 |- ((E.x x = A /\ E.y y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
18		elex 1819	. 2 |- (A e. C -> E.x x = A)
19		elex 1819	. 2 |- (B e. D -> E.y y = B)
20	17, 18, 19	syl2an 454	1 |- ((A e. C /\ B e. D) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  <.cop 2411

This theorem is referenced by:  opelopabg 2817  oprabval6g 4032  ltresr 5258

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-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416

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