Theorem opelopab 4737

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)

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

Allowed substitution hints:   ph( x, y)   ps( 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 4733	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )
6	1, 2, 5	mp2an 683	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1455   e. wcel 1898   _Vcvv 3057   <.cop 3986   {copab 4474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4476

This theorem is referenced by:  opabid2  4983  dfres2  5176  f1oiso  6267  elopabi  6881  xporderlem  6934  cnlnssadj  27782  areacirclem5  32081  dicopelval  34790  dih1dimatlem  34942  pellexlem3  35720