Theorem opelopabaf 4308

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4306 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 4294	. 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 1542	. . . 4 |- F/ x B e. _V
7		opelopabaf.3	. . . 4 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
8	4, 5, 6, 7	sbc2iegf 3060	. . 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 1364   F/wnf 1474   e. wcel 2167   _Vcvv 2763   [.wsbc 2989   <.cop 3625   {copab 4093

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095

This theorem is referenced by: (None)