Theorem nfovd 5973

Description: Deduction version of bound-variable hypothesis builder nfov 5974. (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 5947	. 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 3836	. . 3 ⊢ (𝜑 → Ⅎ𝑥⟨𝐴, 𝐵⟩)
6	2, 5	nffvd 5588	. 2 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘⟨𝐴, 𝐵⟩))
7	1, 6	nfcxfrd 2346	1 ⊢ (𝜑 → Ⅎ𝑥(𝐴𝐹𝐵))

Syntax hints:   → wi 4  Ⅎwnfc 2335  ⟨cop 3636  ‘cfv 5271  (class class class)co 5944

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  nfov  5974  nfnegd  8268