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Metamath Proof Explorer

Theorem syl3anc

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012)

Ref Expression
Hypotheses syl3anc.1 ⊢ ( 𝜑 → 𝜓 )
syl3anc.2 ⊢ ( 𝜑 → 𝜒 )
syl3anc.3 ⊢ ( 𝜑 → 𝜃 )
syl3anc.4 ⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
Assertion syl3anc ( 𝜑 → 𝜏 )

Proof

Step Hyp Ref Expression
1 syl3anc.1 ⊢ ( 𝜑 → 𝜓 )
2 syl3anc.2 ⊢ ( 𝜑 → 𝜒 )
3 syl3anc.3 ⊢ ( 𝜑 → 𝜃 )
4 syl3anc.4 ⊢ ( ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) → 𝜏 )
5 1 2 3 3jca ⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ∧ 𝜃 ) )
6 5 4 syl ⊢ ( 𝜑 → 𝜏 )