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Metamath Proof Explorer

Theorem syl2imc

Description: A commuted version of syl2im . Implication-only version of syl2anr . (Contributed by BJ, 20-Oct-2021)

Ref Expression
Hypotheses syl2im.1 ⊢ ( 𝜑 → 𝜓 )
syl2im.2 ⊢ ( 𝜒 → 𝜃 )
syl2im.3 ⊢ ( 𝜓 → ( 𝜃 → 𝜏 ) )
Assertion syl2imc ( 𝜒 → ( 𝜑 → 𝜏 ) )

Proof

Step Hyp Ref Expression
1 syl2im.1 ⊢ ( 𝜑 → 𝜓 )
2 syl2im.2 ⊢ ( 𝜒 → 𝜃 )
3 syl2im.3 ⊢ ( 𝜓 → ( 𝜃 → 𝜏 ) )
4 1 2 3 syl2im ⊢ ( 𝜑 → ( 𝜒 → 𝜏 ) )
5 4 com12 ⊢ ( 𝜒 → ( 𝜑 → 𝜏 ) )