Theorem opelopabaf 4362

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4360 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 4348	. 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 1574	. . . 4 |- F/ x B e. _V
7		opelopabaf.3	. . . 4 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
8	4, 5, 6, 7	sbc2iegf 3099	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )
9	2, 3, 8	mp2an 426	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> ps )
10	1, 9	bitri 184	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   F/wnf 1506   e. wcel 2200   _Vcvv 2799   [.wsbc 3028   <.cop 3669   {copab 4144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146

This theorem is referenced by: (None)