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Metamath Proof Explorer

Theorem syl2anc

Description: Syllogism inference combined with contraction. (Contributed by NM, 16-Mar-2012)

Ref Expression
Hypotheses syl2anc.1 ⊢ φ → ψ
syl2anc.2 ⊢ φ → χ
syl2anc.3 ⊢ ψ ∧ χ → θ
Assertion syl2anc ⊢ φ → θ

Proof

Step Hyp Ref Expression
1 syl2anc.1 ⊢ φ → ψ
2 syl2anc.2 ⊢ φ → χ
3 syl2anc.3 ⊢ ψ ∧ χ → θ
4 3 ex ⊢ ψ → χ → θ
5 1 2 4 sylc ⊢ φ → θ