Theorem sylan2i 399

Description: A syllogism inference. (Contributed by NM, 1-Aug-1994.)

Hypotheses

Ref	Expression
sylan2i.1	⊢ (𝜑 → 𝜃)
sylan2i.2	⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))

Assertion

Ref	Expression
sylan2i	⊢ (𝜓 → ((𝜒 ∧ 𝜑) → 𝜏))

Proof of Theorem sylan2i

Step	Hyp	Ref	Expression
1		sylan2i.1	. . 3 ⊢ (𝜑 → 𝜃)
2	1	a1i 9	. 2 ⊢ (𝜓 → (𝜑 → 𝜃))
3		sylan2i.2	. 2 ⊢ (𝜓 → ((𝜒 ∧ 𝜃) → 𝜏))
4	2, 3	sylan2d 288	1 ⊢ (𝜓 → ((𝜒 ∧ 𝜑) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  syl2ani  400