Theorem syl3an132 1166

Description: syl2an 596 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)

Hypotheses

Ref	Expression
syl3an132.1	⊢ (𝜑 → 𝜓)
syl3an132.2	⊢ ((𝜒 ∧ 𝜃) → 𝜏)
syl3an132.3	⊢ ((𝜓 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl3an132	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)

Proof of Theorem syl3an132

Step	Hyp	Ref	Expression
1		syl3an132.1	. . 3 ⊢ (𝜑 → 𝜓)
2		syl3an132.2	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜏)
3		syl3an132.3	. . 3 ⊢ ((𝜓 ∧ 𝜏) → 𝜂)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜂)
5	4	3impb 1114	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088

This theorem is referenced by:  2f1fvneq  7252