Theorem eel2131 42223

Description: syl2an 595 with antecedents in standard conjunction form. (Contributed by Alan Sare, 26-Aug-2016.)

Hypotheses

Ref	Expression
eel2131.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
eel2131.2	⊢ ((𝜑 ∧ 𝜃) → 𝜏)
eel2131.3	⊢ ((𝜒 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
eel2131	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 𝜂)

Proof of Theorem eel2131

Step	Hyp	Ref	Expression
1		eel2131.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2		eel2131.2	. . 3 ⊢ ((𝜑 ∧ 𝜃) → 𝜏)
3		eel2131.3	. . 3 ⊢ ((𝜒 ∧ 𝜏) → 𝜂)
4	1, 2, 3	syl2an 595	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜃)) → 𝜂)
5	4	3impdi 1348	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜃) → 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)