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Metamath Proof Explorer

Theorem syl6c

Description: Inference combining syl6 with contraction. (Contributed by Alan Sare, 2-May-2011)

Ref Expression
Hypotheses syl6c.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
syl6c.2 ⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
syl6c.3 ⊢ ( 𝜒 → ( 𝜃 → 𝜏 ) )
Assertion syl6c ( 𝜑 → ( 𝜓 → 𝜏 ) )

Proof

Step Hyp Ref Expression
1 syl6c.1 ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
2 syl6c.2 ⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
3 syl6c.3 ⊢ ( 𝜒 → ( 𝜃 → 𝜏 ) )
4 1 3 syl6 ⊢ ( 𝜑 → ( 𝜓 → ( 𝜃 → 𝜏 ) ) )
5 2 4 mpdd ⊢ ( 𝜑 → ( 𝜓 → 𝜏 ) )