Theorem nfor 1905

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 848	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
2		nf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1858	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4		nf.2	. . 3 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfim 1897	. 2 ⊢ Ⅎ𝑥(¬ 𝜑 → 𝜓)
6	1, 5	nfxfr 1854	1 ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 847  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785

This theorem is referenced by:  nf3or  1906  axi12  2703  axbnd  2704  nfun  4119  nfunOLD  4120  nfpr  4644  rabsnifsb  4674  disjxun  5091  fsuppmapnn0fiubex  13901  nfsum1  15599  nfsum  15600  nfcprod1  15817  nfcprod  15818  fdc1  37807  dvdsrabdioph  42928  mnringmulrcld  44346  disjinfi  45314  iundjiun  46583