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Metamath Proof Explorer

Theorem hban

Description: If x is not free in ph and ps , it is not free in ( ph /\ ps ) . (Contributed by NM, 14-May-1993) (Proof shortened by Wolf Lammen, 2-Jan-2018)

	Ref	Expression
Hypotheses	hb.1	$⊢ φ \to \forall x φ$
	hb.2	$⊢ ψ \to \forall x ψ$
Assertion	hban	$⊢ (φ \land ψ) \to \forall x (φ \land ψ)$

Proof

Step	Hyp	Ref	Expression
1		hb.1	$⊢ φ \to \forall x φ$
2		hb.2	$⊢ ψ \to \forall x ψ$
3	1	nf5i	$⊢ Ⅎ x φ$
4	2	nf5i	$⊢ Ⅎ x ψ$
5	3 4	nfan	$⊢ Ⅎ x (φ \land ψ)$
6	5	nf5ri	$⊢ (φ \land ψ) \to \forall x (φ \land ψ)$