Theorem nfimad 5475

Description: Deduction version of bound-variable hypothesis builder nfima 5474. (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 2770	. . 3 |- F/_ x { z | A. x z e. A }
2		nfaba1 2770	. . 3 |- F/_ x { z | A. x z e. B }
3	1, 2	nfima 5474	. 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 2767	. . . . 5 |- F/ x F/_ x A
7		nfnfc1 2767	. . . . 5 |- F/ x F/_ x B
8	6, 7	nfan 1828	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf 3375	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
10	9	imaeq1d 5465	. . . . 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 3375	. . . . . 6 |- ( F/_ x B -> { z | A. x z e. B } = B )
12	11	imaeq2d 5466	. . . . 5 |- ( F/_ x B -> ( A " { z | A. x z e. B } ) = ( A " B ) )
13	10, 12	sylan9eq 2676	. . . 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 2760	. . 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 693	. 2 |- ( ph -> ( F/_ x ( { z | A. x z e. A } " { z | A. x z e. B } ) <-> F/_ x ( A " B ) ) )
16	3, 15	mpbii 223	1 |- ( ph -> F/_ x ( A " B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   F/_wnfc 2751   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by: (None)