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Metamath Proof Explorer

Theorem mp3an3an

Description: mp3an with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016)

	Ref	Expression
Hypotheses	mp3an3an.1	⊢ 𝜑
	mp3an3an.2	⊢ ( 𝜓 → 𝜒 )
	mp3an3an.3	⊢ ( 𝜃 → 𝜏 )
	mp3an3an.4	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 )
Assertion	mp3an3an	⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜂 )

Proof

Step	Hyp	Ref	Expression
1		mp3an3an.1	⊢ 𝜑
2		mp3an3an.2	⊢ ( 𝜓 → 𝜒 )
3		mp3an3an.3	⊢ ( 𝜃 → 𝜏 )
4		mp3an3an.4	⊢ ( ( 𝜑 ∧ 𝜒 ∧ 𝜏 ) → 𝜂 )
5	1 4	mp3an1	⊢ ( ( 𝜒 ∧ 𝜏 ) → 𝜂 )
6	2 3 5	syl2an	⊢ ( ( 𝜓 ∧ 𝜃 ) → 𝜂 )