Theorem nfovd 7439

Description: Deduction version of bound-variable hypothesis builder nfov 7440. (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 7413	. 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 4855	. . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
6	2, 5	nffvd 6893	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘⟨𝐴, 𝐵⟩))
7	1, 6	nfcxfrd 2924	1 ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵))

Syntax hints:   → wi 4  Ⅎwnfc 2910  ⟨cop 4595  ‘cfv 6536  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is referenced by:  nfov  7440  nfnegd  11447