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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 )