Theorem eel3132 42224

Description: syl2an 595 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)

Hypotheses

Ref	Expression
eel3132.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
eel3132.2	⊢ ((𝜃 ∧ 𝜓) → 𝜏)
eel3132.3	⊢ ((𝜒 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
eel3132	⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜂)

Proof of Theorem eel3132

Step	Hyp	Ref	Expression
1		eel3132.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2		eel3132.2	. . 3 ⊢ ((𝜃 ∧ 𝜓) → 𝜏)
3		eel3132.3	. . 3 ⊢ ((𝜒 ∧ 𝜏) → 𝜂)
4	1, 2, 3	syl2an 595	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜃 ∧ 𝜓)) → 𝜂)
5	4	3impdir 1349	1 ⊢ ((𝜑 ∧ 𝜃 ∧ 𝜓) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085

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 206  df-an 396  df-3an 1087

This theorem is referenced by: (None)