Theorem nfovd 5808

Description: Deduction version of bound-variable hypothesis builder nfov 5809. (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 5785	. 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 3730	. . 3 |- ( ph -> F/_ x <. A , B >. )
6	2, 5	nffvd 5441	. 2 |- ( ph -> F/_ x ( F ` <. A , B >. ) )
7	1, 6	nfcxfrd 2280	1 |- ( ph -> F/_ x ( A F B ) )

Syntax hints:   -> wi 4   F/_wnfc 2269   <.cop 3535   ` cfv 5131  (class class class)co 5782

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  nfov  5809  nfnegd  7982