Theorem nfopd 3850

Description: Deduction version of bound-variable hypothesis builder nfop 3849. This shows how the deduction version of a not-free theorem such as nfop 3849 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 2356	. . 3 |- F/_ x { z | A. x z e. A }
2		nfaba1 2356	. . 3 |- F/_ x { z | A. x z e. B }
3	1, 2	nfop 3849	. 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 2353	. . . . 5 |- F/ x F/_ x A
7		nfnfc1 2353	. . . . 5 |- F/ x F/_ x B
8	6, 7	nfan 1589	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf 2948	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
10	9	adantr 276	. . . . 5 |- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. A } = A )
11		abidnf 2948	. . . . . 6 |- ( F/_ x B -> { z | A. x z e. B } = B )
12	11	adantl 277	. . . . 5 |- ( ( F/_ x A /\ F/_ x B ) -> { z | A. x z e. B } = B )
13	10, 12	opeq12d 3841	. . . 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 2349	. . 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 411	. 2 |- ( ph -> ( F/_ x <. { z | A. x z e. A } , { z | A. x z e. B } >. <-> F/_ x <. A , B >. ) )
16	3, 15	mpbii 148	1 |- ( ph -> F/_ x <. A , B >. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2178   {cab 2193   F/_wnfc 2337   <.cop 3646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652

This theorem is referenced by:  nfbrd  4105  nfovd  5996