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Metamath Proof Explorer

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 )