Theorem opelopabf 4037

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4034 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 4023	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		opelopabf.1	. . 3 |- A e. _V
3		nfcv 2220	. . . . 5 |- F/_ x B
4		opelopabf.x	. . . . 5 |- F/ x ps
5	3, 4	nfsbc 2836	. . . 4 |- F/ x [. B / y ]. ps
6		opelopabf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
7	6	sbcbidv 2873	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
8	5, 7	sbciegf 2846	. . 3 |- ( A e. _V -> ( [. A / x ]. [. B / y ]. ph <-> [. B / y ]. ps ) )
9	2, 8	ax-mp 7	. 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 2846	. . 3 |- ( B e. _V -> ( [. B / y ]. ps <-> ch ) )
14	10, 13	ax-mp 7	. 2 |- ( [. B / y ]. ps <-> ch )
15	1, 9, 14	3bitri 204	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1285   F/wnf 1390   e. wcel 1434   _Vcvv 2602   [.wsbc 2816   <.cop 3409   {copab 3846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-opab 3848

This theorem is referenced by:  pofun  4075  fmptco  5362