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Metamath Proof Explorer

Theorem syl12anc

Description: Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009)

Ref Expression
Hypotheses syl12anc.1 ⊢ ( 𝜑 → 𝜓 )
syl12anc.2 ⊢ ( 𝜑 → 𝜒 )
syl12anc.3 ⊢ ( 𝜑 → 𝜃 )
syl12anc.4 ⊢ ( ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) → 𝜏 )
Assertion syl12anc ( 𝜑 → 𝜏 )

Proof

Step Hyp Ref Expression
1 syl12anc.1 ⊢ ( 𝜑 → 𝜓 )
2 syl12anc.2 ⊢ ( 𝜑 → 𝜒 )
3 syl12anc.3 ⊢ ( 𝜑 → 𝜃 )
4 syl12anc.4 ⊢ ( ( 𝜓 ∧ ( 𝜒 ∧ 𝜃 ) ) → 𝜏 )
5 2 3 jca ⊢ ( 𝜑 → ( 𝜒 ∧ 𝜃 ) )
6 1 5 4 syl2anc ⊢ ( 𝜑 → 𝜏 )