Theorem nfovd 7431

Description: Deduction version of bound-variable hypothesis builder nfov 7432. (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 7405	. 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 4883	. . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
6	2, 5	nffvd 6894	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘⟨𝐴, 𝐵⟩))
7	1, 6	nfcxfrd 2894	1 ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵))

Syntax hints:   → wi 4  Ⅎwnfc 2875  ⟨cop 4627  ‘cfv 6534  (class class class)co 7402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405

This theorem is referenced by:  nfov  7432  nfnegd  11454