Theorem bj-alanim 31781

Description: Closed form of alanimi 1734. (Contributed by BJ, 6-May-2019.)

Assertion

Ref	Expression
bj-alanim	⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))

Proof of Theorem bj-alanim

Step	Hyp	Ref	Expression
1		pm3.3 459	. . . 4 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))
2	1	alimi 1730	. . 3 ⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → ∀𝑥(𝜑 → (𝜓 → 𝜒)))
3		al2im 1732	. . 3 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
4	2, 3	syl 17	. 2 ⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
5	4	impd 446	1 ⊢ (∀𝑥((𝜑 ∧ 𝜓) → 𝜒) → ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒))

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385

This theorem is referenced by: (None)