Theorem opelopabg 2812

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 539	. . 3 |- ((x = A /\ y = B) -> (ph <-> ch))
4	3	copsex2g 2788	. 2 |- ((A e. C /\ B e. D) -> (E.xE.y(<.A, B>. = <.x, y>. /\ ph) <-> ch))
5		elopab 2806	. 2 |- (<.A, B>. e. {<.x, y>. | ph} <-> E.xE.y(<.A, B>. = <.x, y>. /\ ph))
6	4, 5	syl5bb 531	1 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  <.cop 2407  {copab 2661

This theorem is referenced by:  brabg 2813  opelopab2 2814  opelopab 2815  opelcnvg 3291  fvopab3 3768  fvopab3ig 3769  fvopabn 3777  oprabval 4014  brecop 4296  eltopsp 7554  tpsex 7555  istps 7556  ismsg 7750  isring 8093  isvclem 8148  adjt 9796  adjeqt 9798  ishgrag 10641  ispgrag 10651

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-opab 2662

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