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Metamath Proof Explorer

Theorem syl3c

Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 7-Jul-2011)

Ref Expression
Hypotheses syl3c.1 ⊢ ( 𝜑 → 𝜓 )
syl3c.2 ⊢ ( 𝜑 → 𝜒 )
syl3c.3 ⊢ ( 𝜑 → 𝜃 )
syl3c.4 ⊢ ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) )
Assertion syl3c ( 𝜑 → 𝜏 )

Proof

Step Hyp Ref Expression
1 syl3c.1 ⊢ ( 𝜑 → 𝜓 )
2 syl3c.2 ⊢ ( 𝜑 → 𝜒 )
3 syl3c.3 ⊢ ( 𝜑 → 𝜃 )
4 syl3c.4 ⊢ ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) )
5 1 2 4 sylc ⊢ ( 𝜑 → ( 𝜃 → 𝜏 ) )
6 3 5 mpd ⊢ ( 𝜑 → 𝜏 )