Theorem copsex2g 2783

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 1318	. . 3 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
2		hbe1 1012	. . . . 5 |- (E.xE.y(<.A, B>. = <.x, y>. /\ ph) -> A.xE.xE.y(<.A, B>. = <.x, y>. /\ ph))
3		ax-17 968	. . . . 5 |- (ps -> A.xps)
4	2, 3	hbbi 1007	. . . 4 |- ((E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps) -> A.x(E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
5		hbe1 1012	. . . . . . 7 |- (E.y(<.A, B>. = <.x, y>. /\ ph) -> A.yE.y(<.A, B>. = <.x, y>. /\ ph))
6	5	hbex 1003	. . . . . 6 |- (E.xE.y(<.A, B>. = <.x, y>. /\ ph) -> A.yE.xE.y(<.A, B>. = <.x, y>. /\ ph))
7		ax-17 968	. . . . . 6 |- (ps -> A.yps)
8	6, 7	hbbi 1007	. . . . 5 |- ((E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps) -> A.y(E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
9		opeq12 2480	. . . . . . 7 |- ((x = A /\ y = B) -> <.x, y>. = <.A, B>.)
10		copsexg 2782	. . . . . . . 8 |- (<.A, B>. = <.x, y>. -> (ph <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph)))
11	10	eqcoms 1470	. . . . . . 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 528	. . . . 5 |- ((x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
15	8, 14	19.23ai 1060	. . . 4 |- (E.y(x = A /\ y = B) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ps))
16	4, 15	19.23ai 1060	. . 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 1810	. 2 |- (A e. C -> E.x x = A)
19		elex 1810	. 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 953   e. wcel 955  E.wex 977  <.cop 2401

This theorem is referenced by:  opelopabg 2806  oprabval6g 4017  ltresr 5230

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406

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