Theorem nfopd 3730

Description: Deduction version of bound-variable hypothesis builder nfop 3729. This shows how the deduction version of a not-free theorem such as nfop 3729 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 2288	. . 3 |- F/_ x { z | A. x z e. A }
2		nfaba1 2288	. . 3 |- F/_ x { z | A. x z e. B }
3	1, 2	nfop 3729	. 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 2285	. . . . 5 |- F/ x F/_ x A
7		nfnfc1 2285	. . . . 5 |- F/ x F/_ x B
8	6, 7	nfan 1545	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf 2856	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
10	9	adantr 274	. . . . 5 |- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. A } = A )
11		abidnf 2856	. . . . . 6 |- ( F/_ x B -> { z | A. x z e. B } = B )
12	11	adantl 275	. . . . 5 |- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. B } = B )
13	10, 12	opeq12d 3721	. . . 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 2281	. . 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 409	. 2 |- ( ph -> ( F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >. <-> F/_ x <. A , B >. ) )
16	3, 15	mpbii 147	1 |- ( ph -> F/_ x <. A , B >. )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1330   = wceq 1332   e. wcel 1481   {cab 2126   F/_wnfc 2269   <.cop 3535

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541

This theorem is referenced by:  nfbrd  3981  nfovd  5808