Theorem nffvd 5611

Description: Deduction version of bound-variable hypothesis builder nffv 5609. (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 2356	. . 3 |- F/_ x { z | A. x z e. F }
2		nfaba1 2356	. . 3 |- F/_ x { z | A. x z e. A }
3	1, 2	nffv 5609	. 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 2353	. . . . 5 |- F/ x F/_ x F
7		nfnfc1 2353	. . . . 5 |- F/ x F/_ x A
8	6, 7	nfan 1589	. . . 4 |- F/ x ( F/_ x F /\ F/_ x A )
9		abidnf 2948	. . . . . 6 |- ( F/_ x F -> { z | A. x z e. F } = F )
10	9	adantr 276	. . . . 5 |- ( ( F/_ x F /\ F/_ x A ) -> { z | A. x z e. F } = F )
11		abidnf 2948	. . . . . 6 |- ( F/_ x A -> { z | A. x z e. A } = A )
12	11	adantl 277	. . . . 5 |- ( ( F/_ x F /\ F/_ x A ) -> { z | A. x z e. A } = A )
13	10, 12	fveq12d 5606	. . . 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 2349	. . 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 411	. 2 |- ( ph -> ( F/_ x ( { z | A. x z e. F } ` { z | A. x z e. A } ) <-> F/_ x ( F ` A ) ) )
16	3, 15	mpbii 148	1 |- ( ph -> F/_ x ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   e. wcel 2178   {cab 2193   F/_wnfc 2337   ` cfv 5290

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298

This theorem is referenced by:  nfovd  5996  nfixpxy  6827