Theorem nfovd 6057

Description: Deduction version of bound-variable hypothesis builder nfov 6058. (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 6031	. 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 3884	. . 3 |- ( ph -> F/_ x <. A , B >. )
6	2, 5	nffvd 5660	. 2 |- ( ph -> F/_ x ( F ` <. A , B >. ) )
7	1, 6	nfcxfrd 2373	1 |- ( ph -> F/_ x ( A F B ) )

Syntax hints:   -> wi 4   F/_wnfc 2362   <.cop 3676   ` cfv 5333  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  nfov  6058  nfnegd  8434