Theorem opelopab 2816

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

Hypotheses

Ref	Expression
opelopab.1	|- A e. V
opelopab.2	|- B e. V
opelopab.3	|- (x = A -> (ph <-> ps))
opelopab.4	|- (y = B -> (ps <-> ch))

Assertion

Ref	Expression
opelopab	|- (<.A, B>. e. {<.x, y>. | ph} <-> ch)

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

Proof of Theorem opelopab

Step	Hyp	Ref	Expression
1		opelopab.1	. 2 |- A e. V
2		opelopab.2	. 2 |- B e. V
3		opelopab.3	. . 3 |- (x = A -> (ph <-> ps))
4		opelopab.4	. . 3 |- (y = B -> (ps <-> ch))
5	3, 4	opelopabg 2813	. 2 |- ((A e. V /\ B e. V) -> (<.A, B>. e. {<.x, y>. | ph} <-> ch))
6	1, 2, 5	mp2an 696	1 |- (<.A, B>. e. {<.x, y>. | ph} <-> ch)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955   e. wcel 957  Vcvv 1808  <.cop 2408  {copab 2662

This theorem is referenced by:  opabid2 3263  opelco 3284  fnopabfv 3753  f1oiso 3899  tz7.44-1 3923  tz7.44-2 3924  tz7.44-3 3925  elopabi 4110  pw2en 4435  tz9.12lem1 4642  tz9.12lem3 4644  aceq3lem 4715  infmap2lem1 7539  cnlnssadj 9969

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-opab 2663

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