Theorem opelopabgf 4995

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4993 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 4985	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		nfcv 2764	. . . . 5 |- F/_ x B
3		opelopabgf.x	. . . . 5 |- F/ x ps
4	2, 3	nfsbc 3457	. . . 4 |- F/ x [. B / y ]. ps
5		opelopabgf.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
6	5	sbcbidv 3490	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
7	4, 6	sbciegf 3467	. . 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 3467	. . 3 |- ( B e. W -> ( [. B / y ]. ps <-> ch ) )
11	7, 10	sylan9bb 736	. 2 |- ( ( A e. V /\ B e. W ) -> ( [. A / x ]. [. B / y ]. ph <-> ch ) )
12	1, 11	syl5bb 272	1 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   [.wsbc 3435   <.cop 4183   {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713

This theorem is referenced by:  oprabv  6703