Theorem opelopabf 5000

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4997 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 4985	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
2		opelopabf.1	. . 3 |- A e. _V
3		nfcv 2764	. . . . 5 |- F/_ x B
4		opelopabf.x	. . . . 5 |- F/ x ps
5	3, 4	nfsbc 3457	. . . 4 |- F/ x [. B / y ]. ps
6		opelopabf.3	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
7	6	sbcbidv 3490	. . . 4 |- ( x = A -> ( [. B / y ]. ph <-> [. B / y ]. ps ) )
8	5, 7	sbciegf 3467	. . 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 3467	. . 3 |- ( B e. _V -> ( [. B / y ]. ps <-> ch ) )
14	10, 13	ax-mp 5	. 2 |- ( [. B / y ]. ps <-> ch )
15	1, 9, 14	3bitri 286	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> ch )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   F/wnf 1708   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   <.cop 4183   {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713

This theorem is referenced by:  pofun  5051  fmptco  6396  fmptcof2  29457