Theorem nfimad 5032

Description: Deduction version of bound-variable hypothesis builder nfima 5031. (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 2354	. . 3 |- F/_ x { z | A. x z e. A }
2		nfaba1 2354	. . 3 |- F/_ x { z | A. x z e. B }
3	1, 2	nfima 5031	. 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 2351	. . . . 5 |- F/ x F/_ x A
7		nfnfc1 2351	. . . . 5 |- F/ x F/_ x B
8	6, 7	nfan 1588	. . . 4 |- F/ x ( F/_ x A /\ F/_ x B )
9		abidnf 2941	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
10	9	imaeq1d 5022	. . . . 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 2941	. . . . . 6 |- ( F/_ x B -> { z | A. x z e. B } = B )
12	11	imaeq2d 5023	. . . . 5 |- ( F/_ x B -> ( A " { z | A. x z e. B } ) = ( A " B ) )
13	10, 12	sylan9eq 2258	. . . 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 2347	. . 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 1371   e. wcel 2176   {cab 2191   F/_wnfc 2335   "cima 4679

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-cnv 4684  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689

This theorem is referenced by: (None)