Theorem nfovd 5871

Description: Deduction version of bound-variable hypothesis builder nfov 5872. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Hypotheses

Ref	Expression
nfovd.2	|- ( ph -> F/_ x A )
nfovd.3	|- ( ph -> F/_ x F )
nfovd.4	|- ( ph -> F/_ x B )

Assertion

Ref	Expression
nfovd	|- ( ph -> F/_ x ( A F B ) )

Proof of Theorem nfovd

Step	Hyp	Ref	Expression
1		df-ov 5845	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		nfovd.3	. . 3 |- ( ph -> F/_ x F )
3		nfovd.2	. . . 4 |- ( ph -> F/_ x A )
4		nfovd.4	. . . 4 |- ( ph -> F/_ x B )
5	3, 4	nfopd 3775	. . 3 |- ( ph -> F/_ x <. A , B >. )
6	2, 5	nffvd 5498	. 2 |- ( ph -> F/_ x ( F ` <. A , B >. ) )
7	1, 6	nfcxfrd 2306	1 |- ( ph -> F/_ x ( A F B ) )

Syntax hints:   -> wi 4   F/_wnfc 2295   <.cop 3579   ` cfv 5188  (class class class)co 5842

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  nfov  5872  nfnegd  8094