Theorem nfor 1567

Description: If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∨ 𝜓). (Contributed by Jim Kingdon, 11-Mar-2018.)

Hypotheses

Ref	Expression
nfor.1	⊢ Ⅎ𝑥𝜑
nfor.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
nfor	⊢ Ⅎ𝑥(𝜑 ∨ 𝜓)

Proof of Theorem nfor

Step	Hyp	Ref	Expression
1		nfor.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	nfri 1512	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3		nfor.2	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	nfri 1512	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
5	2, 4	hbor 1539	. 2 ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
6	5	nfi 1455	1 ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓)

Syntax hints:   ∨ wo 703  Ⅎwnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-gen 1442  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  nfdc  1652  nfun  3283  nfpr  3633  nfso  4287  nffrec  6375  indpi  7304  nfsum1  11319  nfsum  11320  nfcprod1  11517  nfcprod  11518  bj-findis  14014  isomninnlem  14062  trirec0  14076