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Metamath Proof Explorer

Theorem syl2anc2

Description: Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023)

Ref Expression
Hypotheses syl2anc2.1 ⊢ ( 𝜑 → 𝜓 )
syl2anc2.2 ⊢ ( 𝜓 → 𝜒 )
syl2anc2.3 ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
Assertion syl2anc2 ( 𝜑 → 𝜃 )

Proof

Step Hyp Ref Expression
1 syl2anc2.1 ⊢ ( 𝜑 → 𝜓 )
2 syl2anc2.2 ⊢ ( 𝜓 → 𝜒 )
3 syl2anc2.3 ⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜃 )
4 1 2 syl ⊢ ( 𝜑 → 𝜒 )
5 1 4 3 syl2anc ⊢ ( 𝜑 → 𝜃 )