Theorem sylani 404

Description: A syllogism inference. (Contributed by NM, 2-May-1996.)

Hypotheses

Ref	Expression
sylani.1	⊢ (𝜑 → 𝜒)
sylani.2	⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))

Assertion

Ref	Expression
sylani	⊢ (𝜓 → ((𝜑 ∧ 𝜃) → 𝜏))

Proof of Theorem sylani

Step	Hyp	Ref	Expression
1		sylani.1	. . 3 ⊢ (𝜑 → 𝜒)
2	1	a1i 9	. 2 ⊢ (𝜓 → (𝜑 → 𝜒))
3		sylani.2	. 2 ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))
4	2, 3	syland 291	1 ⊢ (𝜓 → ((𝜑 ∧ 𝜃) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem is referenced by:  syl2ani  406  fiintim  6906  lcmdvds  12033