Theorem nfovd 5947

Description: Deduction version of bound-variable hypothesis builder nfov 5948. (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 5921	. 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 3821	. . 3 |- ( ph -> F/_ x <. A , B >. )
6	2, 5	nffvd 5566	. 2 |- ( ph -> F/_ x ( F ` <. A , B >. ) )
7	1, 6	nfcxfrd 2334	1 |- ( ph -> F/_ x ( A F B ) )

Syntax hints:   -> wi 4   F/_wnfc 2323   <.cop 3621   ` cfv 5254  (class class class)co 5918

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  nfov  5948  nfnegd  8215