Theorem nffvd 5187

Description: Deduction version of bound-variable hypothesis builder nffv 5185. (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 2183	. . 3 |- F/_ x { z | A. x z e. F }
2		nfaba1 2183	. . 3 |- F/_ x { z | A. x z e. A }
3	1, 2	nffv 5185	. 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 2181	. . . . 5 |- F/ x F/_ x F
7		nfnfc1 2181	. . . . 5 |- F/ x F/_ x A
8	6, 7	nfan 1457	. . . 4 |- F/ x ( F/_ x F /\ F/_ x A )
9		abidnf 2709	. . . . . 6 |- ( F/_ x F -> { z | A. x z e. F } = F )
10	9	adantr 261	. . . . 5 |- ( ( F/_ x F /\ F/_ x A ) -> { z | A. x z e. F } = F )
11		abidnf 2709	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
12	11	adantl 262	. . . . 5 |- ( ( F/_ x F /\ F/_ x A ) -> { z | A. x z e. A } = A )
13	10, 12	fveq12d 5184	. . . 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 2177	. . 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 391	. 2 |- ( ph -> ( F/_ x ( { z | A. x z e. F } ` { z | A. x z e. A } ) <-> F/_ x ( F ` A ) ) )
16	3, 15	mpbii 136	1 |- ( ph -> F/_ x ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   F/_wnfc 2165   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910

This theorem is referenced by:  nfovd  5534