Theorem nfovd 6935

Description: Deduction version of bound-variable hypothesis builder nfov 6936. (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 6909	. 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 4641	. . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
6	2, 5	nffvd 6446	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘⟨𝐴, 𝐵⟩))
7	1, 6	nfcxfrd 2969	1 ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵))

Syntax hints:   → wi 4  Ⅎwnfc 2957  ⟨cop 4404  ‘cfv 6124  (class class class)co 6906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-iota 6087  df-fv 6132  df-ov 6909

This theorem is referenced by:  nfov  6936  nfnegd  10597