Theorem nffvd 5481

Description: Deduction version of bound-variable hypothesis builder nffv 5479. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
nffvd	|- ( ph -> F/_ x ( F ` A ) )

Proof of Theorem nffvd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2305	. . 3 |- F/_ x { z | A. x z e. F }
2		nfaba1 2305	. . 3 |- F/_ x { z | A. x z e. A }
3	1, 2	nffv 5479	. 2 |- F/_ x ( { z | A. x z e. F } ` { z | A. x z e. A } )
4		nffvd.2	. . 3 |- ( ph -> F/_ x F )
5		nffvd.3	. . 3 |- ( ph -> F/_ x A )
6		nfnfc1 2302	. . . . 5 |- F/ x F/_ x F
7		nfnfc1 2302	. . . . 5 |- F/ x F/_ x A
8	6, 7	nfan 1545	. . . 4 |- F/ x ( F/_ x F /\ F/_ x A )
9		abidnf 2880	. . . . . 6 |- ( F/_ x F -> { z | A. x z e. F } = F )
10	9	adantr 274	. . . . 5 |- ( ( F/_ x F /\ F/_ x A ) -> { z | A. x z e. F } = F )
11		abidnf 2880	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
12	11	adantl 275	. . . . 5 |- ( ( F/_ x F /\ F/_ x A ) -> { z | A. x z e. A } = A )
13	10, 12	fveq12d 5476	. . . 4 |- ( ( F/_ x F /\ F/_ x A ) -> ( { z | A. x z e. F } ` { z | A. x z e. A } ) = ( F ` A ) )
14	8, 13	nfceqdf 2298	. . 3 |- ( ( F/_ x F /\ F/_ x A ) -> ( F/_ x ( { z | A. x z e. F } ` { z | A. x z e. A } ) <-> F/_ x ( F ` A ) ) )
15	4, 5, 14	syl2anc 409	. 2 |- ( ph -> ( F/_ x ( { z | A. x z e. F } ` { z | A. x z e. A } ) <-> F/_ x ( F ` A ) ) )
16	3, 15	mpbii 147	1 |- ( ph -> F/_ x ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1333   = wceq 1335   e. wcel 2128   {cab 2143   F/_wnfc 2286   ` cfv 5171

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-iota 5136  df-fv 5179

This theorem is referenced by:  nfovd  5851  nfixpxy  6663