Theorem nfor 1927

Description: If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nf.1	⊢ Ⅎ𝑥𝜑
nf.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
nfor	⊢ Ⅎ𝑥(𝜑 ∨ 𝜓)

Proof of Theorem nfor

Step	Hyp	Ref	Expression
1		df-or 861	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
2		nf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1880	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4		nf.2	. . 3 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfim 1919	. 2 ⊢ Ⅎ𝑥(¬ 𝜑 → 𝜓)
6	1, 5	nfxfr 1876	1 ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 860  Ⅎwnf 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1803  df-nf 1807

This theorem is referenced by:  nf3or  1928  axi12  2735  axbnd  2736  nfun  4126  nfpr  4654  rabsnifsb  4684  disjxun  5102  fsuppmapnn0fiubex  14016  nfsum1  15729  nfsum  15730  nfcprod1  15950  nfcprod  15951  fdc1  38252  dvdsrabdioph  43394  mnringmulrcld  44811  disjinfi  45769  iundjiun  47033