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Metamath Proof Explorer

Theorem syl13anc

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 )
syl3Xanc.4 ⊢ ( 𝜑 → 𝜏 )
syl13anc.5 ⊢ ( ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )
Assertion syl13anc ( 𝜑 → 𝜂 )

Proof

Step Hyp Ref Expression
1 syl3anc.1 ⊢ ( 𝜑 → 𝜓 )
2 syl3anc.2 ⊢ ( 𝜑 → 𝜒 )
3 syl3anc.3 ⊢ ( 𝜑 → 𝜃 )
4 syl3Xanc.4 ⊢ ( 𝜑 → 𝜏 )
5 syl13anc.5 ⊢ ( ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) ) → 𝜂 )
6 2 3 4 3jca ⊢ ( 𝜑 → ( 𝜒 ∧ 𝜃 ∧ 𝜏 ) )
7 1 6 5 syl2anc ⊢ ( 𝜑 → 𝜂 )