Theorem nffvd 5218

Description: Deduction version of bound-variable hypothesis builder nffv 5216. (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 2225	. . 3 |- F/_ x { z | A. x z e. F }
2		nfaba1 2225	. . 3 |- F/_ x { z | A. x z e. A }
3	1, 2	nffv 5216	. 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 2223	. . . . 5 |- F/ x F/_ x F
7		nfnfc1 2223	. . . . 5 |- F/ x F/_ x A
8	6, 7	nfan 1498	. . . 4 |- F/ x ( F/_ x F /\ F/_ x A )
9		abidnf 2761	. . . . . 6 |- ( F/_ x F -> { z | A. x z e. F } = F )
10	9	adantr 270	. . . . 5 |- ( ( F/_ x F /\ F/_ x A ) -> { z | A. x z e. F } = F )
11		abidnf 2761	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
12	11	adantl 271	. . . . 5 |- ( ( F/_ x F /\ F/_ x A ) -> { z | A. x z e. A } = A )
13	10, 12	fveq12d 5215	. . . 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 2219	. . 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 403	. 2 |- ( ph -> ( F/_ x ( { z | A. x z e. F } ` { z | A. x z e. A } ) <-> F/_ x ( F ` A ) ) )
16	3, 15	mpbii 146	1 |- ( ph -> F/_ x ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   = wceq 1285   e. wcel 1434   {cab 2068   F/_wnfc 2207   ` cfv 4932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940

This theorem is referenced by:  nfovd  5565