Theorem alanimi

Description: Variant of al2imi with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011)

	Ref	Expression
Hypothesis	alanimi.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
Assertion	alanimi	⊢ ( ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) → ∀ 𝑥 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		alanimi.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
2	1	ex	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
3	2	al2imi	⊢ ( ∀ 𝑥 𝜑 → ( ∀ 𝑥 𝜓 → ∀ 𝑥 𝜒 ) )
4	3	imp	⊢ ( ( ∀ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) → ∀ 𝑥 𝜒 )