Theorem nfor 1937

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 862	. 2 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
2		nf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1890	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4		nf.2	. . 3 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfim 1929	. 2 ⊢ Ⅎ𝑥(¬ 𝜑 → 𝜓)
6	1, 5	nfxfr 1886	1 ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 861  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  nf3or  1938  axi12  2736  axbnd  2737  nfun  4127  nfpr  4663  rabsnifsb  4693  disjxun  5112  fsuppmapnn0fiubex  14048  nfsum1  15767  nfsum  15768  nfcprod1  15988  nfcprod  15989  fdc1  38438  dvdsrabdioph  43578  mnringmulrcld  44993  disjinfi  45951  iundjiun  47215