Theorem nffvd 5529

Description: Deduction version of bound-variable hypothesis builder nffv 5527. (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 2325	. . 3 |- F/_ x { z | A. x z e. F }
2		nfaba1 2325	. . 3 |- F/_ x { z | A. x z e. A }
3	1, 2	nffv 5527	. 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 2322	. . . . 5 |- F/ x F/_ x F
7		nfnfc1 2322	. . . . 5 |- F/ x F/_ x A
8	6, 7	nfan 1565	. . . 4 |- F/ x ( F/_ x F /\ F/_ x A )
9		abidnf 2907	. . . . . 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 2907	. . . . . 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 5524	. . . 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 2318	. . 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 1351   = wceq 1353   e. wcel 2148   {cab 2163   F/_wnfc 2306   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226

This theorem is referenced by:  nfovd  5906  nfixpxy  6719