Theorem opelopabgf 4304

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4302 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.)

Hypotheses

Ref	Expression
opelopabgf.x	|- F/ x ps
opelopabgf.y	|- F/ y ch
opelopabgf.1	|- ( x = A -> ( ph <-> ps ) )
opelopabgf.2	|- ( y = B -> ( ps <-> ch ) )

Assertion

Ref	Expression
opelopabgf	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )

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

Allowed substitution hints:   ph( x, y)   ps( x, y)   ch( x, y)   V( x, y)   W( x, y)

Proof of Theorem opelopabgf

Step	Hyp	Ref	Expression
1		opelopabsb 4294	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		nfcv 2339	. . . . 5 |- F/_ x B
3		opelopabgf.x	. . . . 5 |- F/ x ps
4	2, 3	nfsbcw 3119	. . . 4 |- F/ x [. B / y ]. ps
5		opelopabgf.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
6	5	sbcbidv 3048	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
7	4, 6	sbciegf 3021	. . 3 |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. ps ) )
8		opelopabgf.y	. . . 4 |- F/ y ch
9		opelopabgf.2	. . . 4 |- ( y = B -> ( ps <-> ch ) )
10	8, 9	sbciegf 3021	. . 3 |- ( B e. W -> ( [. B / y ]. ps <-> ch ) )
11	7, 10	sylan9bb 462	. 2 |- ( ( A e. V /\ B e. W ) -> ( [. A / x ]. [. B / y ]. ph <-> ch ) )
12	1, 11	bitrid 192	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1474   e. wcel 2167   [.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:  opabfi  6999