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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 ∨ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nf.1	⊢ Ⅎ 𝑥 𝜑
2		nf.2	⊢ Ⅎ 𝑥 𝜓
3		df-or	⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( ¬ 𝜑 → 𝜓 ) )
4	1	nfn	⊢ Ⅎ 𝑥 ¬ 𝜑
5	4 2	nfim	⊢ Ⅎ 𝑥 ( ¬ 𝜑 → 𝜓 )
6	3 5	nfxfr	⊢ Ⅎ 𝑥 ( 𝜑 ∨ 𝜓 )