Theorem opelopabaf 4251

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4249 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
opelopabaf.x	|- F/ x ps
opelopabaf.y	|- F/ y ps
opelopabaf.1	|- A e. _V
opelopabaf.2	|- B e. _V
opelopabaf.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )

Assertion

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

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

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem opelopabaf

Step	Hyp	Ref	Expression
1		opelopabsb 4238	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		opelopabaf.1	. . 3 |- A e. _V
3		opelopabaf.2	. . 3 |- B e. _V
4		opelopabaf.x	. . . 4 |- F/ x ps
5		opelopabaf.y	. . . 4 |- F/ y ps
6		nfv 1516	. . . 4 |- F/ x B e. _V
7		opelopabaf.3	. . . 4 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
8	4, 5, 6, 7	sbc2iegf 3021	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )
9	2, 3, 8	mp2an 423	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> ps )
10	1, 9	bitri 183	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   F/wnf 1448   e. wcel 2136   _Vcvv 2726   [.wsbc 2951   <.cop 3579   {copab 4042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044

This theorem is referenced by: (None)