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Metamath Proof Explorer

Theorem syl2an

Description: A double syllogism inference. For an implication-only version, see syl2im . (Contributed by NM, 31-Jan-1997)

Ref Expression
Hypotheses syl2an.1 ⊢ ( 𝜑 → 𝜓 )
syl2an.2 ⊢ ( 𝜏 → 𝜒 )
syl2an.3 ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion syl2an ( ( 𝜑 ∧ 𝜏 ) → 𝜃 )

Proof

Step Hyp Ref Expression
1 syl2an.1 ⊢ ( 𝜑 → 𝜓 )
2 syl2an.2 ⊢ ( 𝜏 → 𝜒 )
3 syl2an.3 ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
4 1 3 sylan ⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
5 2 4 sylan2 ⊢ ( ( 𝜑 ∧ 𝜏 ) → 𝜃 )