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Metamath Proof Explorer

Theorem syl22anc

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

Ref Expression
Hypotheses syl12anc.1 ⊢ ( 𝜑 → 𝜓 )
syl12anc.2 ⊢ ( 𝜑 → 𝜒 )
syl12anc.3 ⊢ ( 𝜑 → 𝜃 )
syl22anc.4 ⊢ ( 𝜑 → 𝜏 )
syl22anc.5 ⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ) ) → 𝜂 )
Assertion syl22anc ( 𝜑 → 𝜂 )

Proof

Step Hyp Ref Expression
1 syl12anc.1 ⊢ ( 𝜑 → 𝜓 )
2 syl12anc.2 ⊢ ( 𝜑 → 𝜒 )
3 syl12anc.3 ⊢ ( 𝜑 → 𝜃 )
4 syl22anc.4 ⊢ ( 𝜑 → 𝜏 )
5 syl22anc.5 ⊢ ( ( ( 𝜓 ∧ 𝜒 ) ∧ ( 𝜃 ∧ 𝜏 ) ) → 𝜂 )
6 1 2 jca ⊢ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) )
7 6 3 4 5 syl12anc ⊢ ( 𝜑 → 𝜂 )