Theorem opelopabgf 4300

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4298 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 4290	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		nfcv 2336	. . . . 5 |- F/_ x B
3		opelopabgf.x	. . . . 5 |- F/ x ps
4	2, 3	nfsbcw 3115	. . . 4 |- F/ x [. B / y ]. ps
5		opelopabgf.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
6	5	sbcbidv 3044	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
7	4, 6	sbciegf 3017	. . 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 3017	. . 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 1471   e. wcel 2164   [.wsbc 2985   <.cop 3621   {copab 4089

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091

This theorem is referenced by:  opabfi  6992