Theorem nfopd 3782

Description: Deduction version of bound-variable hypothesis builder nfop 3781. This shows how the deduction version of a not-free theorem such as nfop 3781 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 2318	. . 3 |- F/_ x { z | A. x z e. A }
2		nfaba1 2318	. . 3 |- F/_ x { z | A. x z e. B }
3	1, 2	nfop 3781	. 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 2315	. . . . 5 |- F/ x F/_ x A
7		nfnfc1 2315	. . . . 5 |- F/ x F/_ x B
8	6, 7	nfan 1558	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf 2898	. . . . . 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 2898	. . . . . 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 3773	. . . 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 2311	. . 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 1346   = wceq 1348   e. wcel 2141   {cab 2156   F/_wnfc 2299   <.cop 3586

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592

This theorem is referenced by:  nfbrd  4034  nfovd  5882