Theorem opelopabgf 4407

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4405 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 4397	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		nfcv 2392	. . . . 5 |- F/_ x B
3		opelopabgf.x	. . . . 5 |- F/ x ps
4	2, 3	nfsbcw 3182	. . . 4 |- F/ x [. B / y ]. ps
5		opelopabgf.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
6	5	sbcbidv 3110	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
7	4, 6	sbciegf 3083	. . 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 3083	. . 3 |- ( B e. W -> ( [. B / y ]. ps <-> ch ) )
11	7, 10	sylan9bb 466	. 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 1402   F/wnf 1513   e. wcel 2209   [.wsbc 3051   <.cop 3708   {copab 4186

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188

This theorem is referenced by:  opabfi  7237