Theorem opelopabf 4375

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4372 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 4360	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		opelopabf.1	. . 3 |- A e. _V
3		nfcv 2375	. . . . 5 |- F/_ x B
4		opelopabf.x	. . . . 5 |- F/ x ps
5	3, 4	nfsbc 3053	. . . 4 |- F/ x [. B / y ]. ps
6		opelopabf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
7	6	sbcbidv 3091	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
8	5, 7	sbciegf 3064	. . 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 3064	. . 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 1398   F/wnf 1509   e. wcel 2202   _Vcvv 2803   [.wsbc 3032   <.cop 3676   {copab 4154

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156

This theorem is referenced by:  pofun  4415  fmptco  5821  uchoice  6309