Theorem opelopab2 2819

Description: Ordered pair membership in an ordered pair class abstraction.

Hypotheses

Ref	Expression
opelopab2.1	|- (x = A -> (ph <-> ps))
opelopab2.2	|- (y = B -> (ps <-> ch))

Assertion

Ref	Expression
opelopab2	|- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ((x e. C /\ y e. D) /\ ph)} <-> ch))

Distinct variable groups:   x,y,A   x,B,y   x,C,y   x,D,y   ch,x,y

Proof of Theorem opelopab2

Step	Hyp	Ref	Expression
1		eleq1 1534	. . . . 5 |- (x = A -> (x e. C <-> A e. C))
2	1	anbi1d 617	. . . 4 |- (x = A -> ((x e. C /\ y e. D) <-> (A e. C /\ y e. D)))
3		opelopab2.1	. . . 4 |- (x = A -> (ph <-> ps))
4	2, 3	anbi12d 628	. . 3 |- (x = A -> (((x e. C /\ y e. D) /\ ph) <-> ((A e. C /\ y e. D) /\ ps)))
5		eleq1 1534	. . . . 5 |- (y = B -> (y e. D <-> B e. D))
6	5	anbi2d 616	. . . 4 |- (y = B -> ((A e. C /\ y e. D) <-> (A e. C /\ B e. D)))
7		opelopab2.2	. . . 4 |- (y = B -> (ps <-> ch))
8	6, 7	anbi12d 628	. . 3 |- (y = B -> (((A e. C /\ y e. D) /\ ps) <-> ((A e. C /\ B e. D) /\ ch)))
9	4, 8	opelopabg 2817	. 2 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ((x e. C /\ y e. D) /\ ph)} <-> ((A e. C /\ B e. D) /\ ch)))
10		pm3.27 323	. . 3 |- (((A e. C /\ B e. D) /\ ch) -> ch)
11		pm3.2 283	. . 3 |- ((A e. C /\ B e. D) -> (ch -> ((A e. C /\ B e. D) /\ ch)))
12	10, 11	impbid2 518	. 2 |- ((A e. C /\ B e. D) -> (((A e. C /\ B e. D) /\ ch) <-> ch))
13	9, 12	bitrd 528	1 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ((x e. C /\ y e. D) /\ ph)} <-> ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  <.cop 2411  {copab 2666

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  df-opab 2667

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