Theorem syl3anl2 1184

Description: A syllogism inference. (Contributed by NM, 24-Feb-2005.)

Hypotheses

Ref	Expression
syl3anl2.1	⊢ (𝜑 → 𝜒)
syl3anl2.2	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl3anl2	⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Proof of Theorem syl3anl2

Step	Hyp	Ref	Expression
1		syl3anl2.1	. . 3 ⊢ (𝜑 → 𝜒)
2		syl3anl2.2	. . . 4 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)
3	2	ex 108	. . 3 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜃) → (𝜏 → 𝜂))
4	1, 3	syl3an2 1169	. 2 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜃) → (𝜏 → 𝜂))
5	4	imp 115	1 ⊢ (((𝜓 ∧ 𝜑 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 887

This theorem is referenced by:  syl3anr2  1188