Theorem nfimad 4697

Description: Deduction version of bound-variable hypothesis builder nfima 4696. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfimad	|- ( ph -> F/_ x ( A " B ) )

Proof of Theorem nfimad

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2224	. . 3 |- F/_ x { z | A. x z e. A }
2		nfaba1 2224	. . 3 |- F/_ x { z | A. x z e. B }
3	1, 2	nfima 4696	. 2 |- F/_ x ( { z | A. x z e. A } " { z | A. x z e. B } )
4		nfimad.2	. . 3 |- ( ph -> F/_ x A )
5		nfimad.3	. . 3 |- ( ph -> F/_ x B )
6		nfnfc1 2222	. . . . 5 |- F/ x F/_ x A
7		nfnfc1 2222	. . . . 5 |- F/ x F/_ x B
8	6, 7	nfan 1497	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf 2760	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
10	9	imaeq1d 4687	. . . . 5 |- ( F/_ x A -> ( { z | A. x z e. A } " { z | A. x z e. B } ) = ( A " { z | A. x z e. B } ) )
11		abidnf 2760	. . . . . 6 |- ( F/_ x B -> { z | A. x z e. B } = B )
12	11	imaeq2d 4688	. . . . 5 |- ( F/_ x B -> ( A " { z | A. x z e. B } ) = ( A " B ) )
13	10, 12	sylan9eq 2133	. . . 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 2218	. . 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 403	. 2 |- ( ph -> ( F/_ x ( { z | A. x z e. A } " { z | A. x z e. B } ) <-> F/_ x ( A " B ) ) )
16	3, 15	mpbii 146	1 |- ( ph -> F/_ x ( A " B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   e. wcel 1433   {cab 2067   F/_wnfc 2206   "cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by: (None)