Theorem eel132 41029

Description: syl2an 597 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)

Hypotheses

Ref	Expression
eel132.1	⊢ (𝜑 → 𝜓)
eel132.2	⊢ ((𝜒 ∧ 𝜃) → 𝜏)
eel132.3	⊢ ((𝜓 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
eel132	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)

Proof of Theorem eel132

Step	Hyp	Ref	Expression
1		eel132.1	. . 3 ⊢ (𝜑 → 𝜓)
2		eel132.2	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜏)
3		eel132.3	. . 3 ⊢ ((𝜓 ∧ 𝜏) → 𝜂)
4	1, 2, 3	syl2an 597	. 2 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜂)
5	4	3impb 1111	1 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083

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 209  df-an 399  df-3an 1085

This theorem is referenced by: (None)