Theorem opelopabg 2817

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61.

Hypotheses

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

Assertion

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

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

Proof of Theorem opelopabg

Step	Hyp	Ref	Expression
1		opelopabg.1	. . . 4 |- (x = A -> (ph <-> ps))
2		opelopabg.2	. . . 4 |- (y = B -> (ps <-> ch))
3	1, 2	sylan9bb 540	. . 3 |- ((x = A /\ y = B) -> (ph <-> ch))
4	3	copsex2g 2793	. 2 |- ((A e. C /\ B e. D) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ch))
5		elopab 2811	. 2 |- (<.A, B>. e. {<.x, y>. | ph} <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph))
6	4, 5	syl5bb 532	1 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  <.cop 2411  {copab 2666

This theorem is referenced by:  brabg 2818  opelopab2 2819  opelopab 2820  opelcnvg 3296  fvopab3 3777  fvopab3ig 3778  fvopabn 3786  oprabval 4023  brecop 4306  eltopsp 7604  tpsex 7605  istps 7606  ismsg 7800  isring 8141  isvclem 8196  adjt 9857  adjeqt 9859  ishgrag 10769  ispgrag 10779

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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