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Metamath Proof Explorer

Theorem nfor

Description: If x is not free in ph and ps , then it is not free in ( ph \/ ps ) . (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 11-Aug-2016)

	Ref	Expression
Hypotheses	nf.1	$⊢ Ⅎ x φ$
	nf.2	$⊢ Ⅎ x ψ$
Assertion	nfor	$⊢ Ⅎ x (φ \lor ψ)$

Proof

Step	Hyp	Ref	Expression
1		nf.1	$⊢ Ⅎ x φ$
2		nf.2	$⊢ Ⅎ x ψ$
3		df-or	$⊢ (φ \lor ψ) \leftrightarrow (\neg φ \to ψ)$
4	1	nfn	$⊢ Ⅎ x \neg φ$
5	4 2	nfim	$⊢ Ⅎ x (\neg φ \to ψ)$
6	3 5	nfxfr	$⊢ Ⅎ x (φ \lor ψ)$