Theorem nfovd 5554

Description: Deduction version of bound-variable hypothesis builder nfov 5555. (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 5535	. 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 3587	. . 3 |- ( ph -> F/_ x <. A , B >. )
6	2, 5	nffvd 5207	. 2 |- ( ph -> F/_ x ( F ` <. A , B >. ) )
7	1, 6	nfcxfrd 2217	1 |- ( ph -> F/_ x ( A F B ) )

Syntax hints:   -> wi 4   F/_wnfc 2206   <.cop 3401   ` cfv 4922  (class class class)co 5532

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  nfov  5555  nfnegd  7304