Theorem opelopabf 4252

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4249 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 4238	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		opelopabf.1	. . 3 |- A e. _V
3		nfcv 2308	. . . . 5 |- F/_ x B
4		opelopabf.x	. . . . 5 |- F/ x ps
5	3, 4	nfsbc 2971	. . . 4 |- F/ x [. B / y ]. ps
6		opelopabf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
7	6	sbcbidv 3009	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
8	5, 7	sbciegf 2982	. . 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 2982	. . 3 |- ( B e. _V -> ( [. B / y ]. ps <-> ch ) )
14	10, 13	ax-mp 5	. 2 |- ( [. B / y ]. ps <-> ch )
15	1, 9, 14	3bitri 205	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   F/wnf 1448   e. wcel 2136   _Vcvv 2726   [.wsbc 2951   <.cop 3579   {copab 4042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044

This theorem is referenced by:  pofun  4290  fmptco  5651