Theorem opelopabgf 4357

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4355 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 4347	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		nfcv 2372	. . . . 5 |- F/_ x B
3		opelopabgf.x	. . . . 5 |- F/ x ps
4	2, 3	nfsbcw 3159	. . . 4 |- F/ x [. B / y ]. ps
5		opelopabgf.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
6	5	sbcbidv 3087	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
7	4, 6	sbciegf 3060	. . 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 3060	. . 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 1395   F/wnf 1506   e. wcel 2200   [.wsbc 3028   <.cop 3669   {copab 4143

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 4201  ax-pow 4257  ax-pr 4292

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 4145

This theorem is referenced by:  opabfi  7088