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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 → ∀ 𝑥 𝜑 )
	hb.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
Assertion	hban	⊢ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		hb.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		hb.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3	1	nf5i	⊢ Ⅎ 𝑥 𝜑
4	2	nf5i	⊢ Ⅎ 𝑥 𝜓
5	3 4	nfan	⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝜓 )
6	5	nf5ri	⊢ ( ( 𝜑 ∧ 𝜓 ) → ∀ 𝑥 ( 𝜑 ∧ 𝜓 ) )