Theorem nfopd 3821

Description: Deduction version of bound-variable hypothesis builder nfop 3820. This shows how the deduction version of a not-free theorem such as nfop 3820 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 2342	. . 3 |- F/_ x { z | A. x z e. A }
2		nfaba1 2342	. . 3 |- F/_ x { z | A. x z e. B }
3	1, 2	nfop 3820	. 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 2339	. . . . 5 |- F/ x F/_ x A
7		nfnfc1 2339	. . . . 5 |- F/ x F/_ x B
8	6, 7	nfan 1576	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf 2928	. . . . . 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 2928	. . . . . 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 3812	. . . 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 2335	. . 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 1362   = wceq 1364   e. wcel 2164   {cab 2179   F/_wnfc 2323   <.cop 3621

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627

This theorem is referenced by:  nfbrd  4074  nfovd  5947