Theorem copsex2g 1903

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 group(s):   x,y,ps   x,A,y   x,B,y

Proof of Theorem copsex2g

Step	Hyp	Ref	Expression
1		eeanv 980	. . 3 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
2		hbe1 709	. . . . 5 |- (E.xE.y(<.A, B>. = <.x, y>. /\ ph) -> A.xE.xE.y(<.A, B>. = <.x, y>. /\ ph))
3		ax-17 925	. . . . 5 |- (ps -> A.xps)
4	2, 3	hbbi 705	. . . 4 |- ((E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps) -> A.x(E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
5		hbe1 709	. . . . . . 7 |- (E.y(<.A, B>. = <.x, y>. /\ ph) -> A.yE.y(<.A, B>. = <.x, y>. /\ ph))
6	5	hbex 701	. . . . . 6 |- (E.xE.y(<.A, B>. = <.x, y>. /\ ph) -> A.yE.xE.y(<.A, B>. = <.x, y>. /\ ph))
7		ax-17 925	. . . . . 6 |- (ps -> A.yps)
8	6, 7	hbbi 705	. . . . 5 |- ((E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps) -> A.y(E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
9		opeq12 1878	. . . . . . 7 |- ((x = A /\ y = B) -> <.x, y>. = <.A, B>.)
10		copsexg 1902	. . . . . . . 8 |- (<.A, B>. = <.x, y>. -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
11	10	cleqcoms 1104	. . . . . . 7 |- (<.x, y>. = <.A, B>. -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
12	9, 11	syl 12	. . . . . 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 408	. . . . 5 |- ((x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
15	8, 14	19.23ai 746	. . . 4 |- (E.y(x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
16	4, 15	19.23ai 746	. . 3 |- (E.xE.y(x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
17	1, 16	sylbir 176	. 2 |- ((E.x x = A /\ E.y y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
18		elex 1356	. 2 |- (A e. C -> E.x x = A)
19		elex 1356	. 2 |- (B e. D -> E.y y = B)
20	17, 18, 19	syl2an 349	1 |- ((A e. C /\ B e. D) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  <.cop 1810

This theorem is referenced by:  ltresr 4052

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815

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