Theorem opelopabf 4369

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4366 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008.)

Hypotheses

Ref	Expression
opelopabf.x	|- F/ x ps
opelopabf.y	|- F/ y ch
opelopabf.1	|- A e. _V
opelopabf.2	|- B e. _V
opelopabf.3	|- ( x = A -> ( ph <-> ps ) )
opelopabf.4	|- ( y = B -> ( ps <-> ch ) )

Assertion

Ref	Expression
opelopabf	|- ( <. 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)

Proof of Theorem opelopabf

Step	Hyp	Ref	Expression
1		opelopabsb 4354	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		opelopabf.1	. . 3 |- A e. _V
3		nfcv 2374	. . . . 5 |- F/_ x B
4		opelopabf.x	. . . . 5 |- F/ x ps
5	3, 4	nfsbc 3052	. . . 4 |- F/ x [. B / y ]. ps
6		opelopabf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
7	6	sbcbidv 3090	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
8	5, 7	sbciegf 3063	. . 3 |- ( A e. _V -> ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. ps ) )
9	2, 8	ax-mp 5	. 2 |- ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. ps )
10		opelopabf.2	. . 3 |- B e. _V
11		opelopabf.y	. . . 4 |- F/ y ch
12		opelopabf.4	. . . 4 |- ( y = B -> ( ps <-> ch ) )
13	11, 12	sbciegf 3063	. . 3 |- ( B e. _V -> ( [. B / y ]. ps <-> ch ) )
14	10, 13	ax-mp 5	. 2 |- ( [. B / y ]. ps <-> ch )
15	1, 9, 14	3bitri 206	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   F/wnf 1508   e. wcel 2202   _Vcvv 2802   [.wsbc 3031   <.cop 3672   {copab 4149

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151

This theorem is referenced by:  pofun  4409  fmptco  5813  uchoice  6299