Theorem syl3an132

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

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

Proof

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