Theorem nfimad 5115

Description: Deduction version of bound-variable hypothesis builder nfima 5114. (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 2392	. . 3 |- F/_ x { z | A. x z e. A }
2		nfaba1 2392	. . 3 |- F/_ x { z | A. x z e. B }
3	1, 2	nfima 5114	. 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 2389	. . . . 5 |- F/ x F/_ x A
7		nfnfc1 2389	. . . . 5 |- F/ x F/_ x B
8	6, 7	nfan 1614	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf 2988	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
10	9	imaeq1d 5105	. . . . 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 2988	. . . . . 6 |- ( F/_ x B -> { z | A. x z e. B } = B )
12	11	imaeq2d 5106	. . . . 5 |- ( F/_ x B -> ( A " { z | A. x z e. B } ) = ( A " B ) )
13	10, 12	sylan9eq 2287	. . . 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 2385	. . 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 1396   e. wcel 2205   {cab 2220   F/_wnfc 2373   "cima 4757

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rab 2531  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767

This theorem is referenced by: (None)