Metamath Proof Explorer


Theorem ee333

Description: e333 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ee333.1 ( 𝜑 → ( 𝜓 → ( 𝜒𝜃 ) ) )
ee333.2 ( 𝜑 → ( 𝜓 → ( 𝜒𝜏 ) ) )
ee333.3 ( 𝜑 → ( 𝜓 → ( 𝜒𝜂 ) ) )
ee333.4 ( 𝜃 → ( 𝜏 → ( 𝜂𝜁 ) ) )
Assertion ee333 ( 𝜑 → ( 𝜓 → ( 𝜒𝜁 ) ) )

Proof

Step Hyp Ref Expression
1 ee333.1 ( 𝜑 → ( 𝜓 → ( 𝜒𝜃 ) ) )
2 ee333.2 ( 𝜑 → ( 𝜓 → ( 𝜒𝜏 ) ) )
3 ee333.3 ( 𝜑 → ( 𝜓 → ( 𝜒𝜂 ) ) )
4 ee333.4 ( 𝜃 → ( 𝜏 → ( 𝜂𝜁 ) ) )
5 1 3imp ( ( 𝜑𝜓𝜒 ) → 𝜃 )
6 2 3imp ( ( 𝜑𝜓𝜒 ) → 𝜏 )
7 3 3imp ( ( 𝜑𝜓𝜒 ) → 𝜂 )
8 5 6 7 4 syl3c ( ( 𝜑𝜓𝜒 ) → 𝜁 )
9 8 3exp ( 𝜑 → ( 𝜓 → ( 𝜒𝜁 ) ) )