Theorem nfovd 7449

Description: Deduction version of bound-variable hypothesis builder nfov 7450. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Hypotheses

Ref	Expression
nfovd.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfovd.3	⊢ (𝜑 → Ⅎ𝑥𝐹)
nfovd.4	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfovd	⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵))

Proof of Theorem nfovd

Step	Hyp	Ref	Expression
1		df-ov 7423	. 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		nfovd.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐹)
3		nfovd.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
4		nfovd.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
5	3, 4	nfopd 4891	. . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
6	2, 5	nffvd 6909	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘⟨𝐴, 𝐵⟩))
7	1, 6	nfcxfrd 2898	1 ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵))

Syntax hints:   → wi 4  Ⅎwnfc 2879  ⟨cop 4635  ‘cfv 6548  (class class class)co 7420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423

This theorem is referenced by:  nfov  7450  nfnegd  11485