Theorem eel3132

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

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

Proof

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