Theorem nfopd 4419

Description: Deduction version of bound-variable hypothesis builder nfop 4418. This shows how the deduction version of a not-free theorem such as nfop 4418 can be created from the corresponding not-free inference theorem. (Contributed by NM, 4-Feb-2008.)

Hypotheses

Ref	Expression
nfopd.2	|- ( ph -> F/_ x A )
nfopd.3	|- ( ph -> F/_ x B )

Assertion

Ref	Expression
nfopd	|- ( ph -> F/_ x <. A , B >. )

Proof of Theorem nfopd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2770	. . 3 |- F/_ x { z | A. x z e. A }
2		nfaba1 2770	. . 3 |- F/_ x { z | A. x z e. B }
3	1, 2	nfop 4418	. 2 |- F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >.
4		nfopd.2	. . 3 |- ( ph -> F/_ x A )
5		nfopd.3	. . 3 |- ( ph -> F/_ x B )
6		nfnfc1 2767	. . . . 5 |- F/ x F/_ x A
7		nfnfc1 2767	. . . . 5 |- F/ x F/_ x B
8	6, 7	nfan 1828	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf 3375	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
10	9	adantr 481	. . . . 5 |- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. A } = A )
11		abidnf 3375	. . . . . 6 |- ( F/_ x B -> { z | A. x z e. B } = B )
12	11	adantl 482	. . . . 5 |- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. B } = B )
13	10, 12	opeq12d 4410	. . . 4 |- ( ( F/_ x A /\ F/_ x B ) -> <. { z | A. x z e. A } , { z | A. x z e. B } >. = <. A , B >. )
14	8, 13	nfceqdf 2760	. . 3 |- ( ( F/_ x A /\ F/_ x B ) -> ( F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >. <-> F/_ x <. A , B >. ) )
15	4, 5, 14	syl2anc 693	. 2 |- ( ph -> ( F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >. <-> F/_ x <. A , B >. ) )
16	3, 15	mpbii 223	1 |- ( ph -> F/_ x <. A , B >. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   F/_wnfc 2751   <.cop 4183

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  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

This theorem is referenced by:  nfbrd  4698  dfid3  5025  nfovd  6675