Metamath Proof Explorer


Theorem ee112

Description: e112 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ee112.1 ( 𝜑𝜓 )
ee112.2 ( 𝜑𝜒 )
ee112.3 ( 𝜑 → ( 𝜃𝜏 ) )
ee112.4 ( 𝜓 → ( 𝜒 → ( 𝜏𝜂 ) ) )
Assertion ee112 ( 𝜑 → ( 𝜃𝜂 ) )

Proof

Step Hyp Ref Expression
1 ee112.1 ( 𝜑𝜓 )
2 ee112.2 ( 𝜑𝜒 )
3 ee112.3 ( 𝜑 → ( 𝜃𝜏 ) )
4 ee112.4 ( 𝜓 → ( 𝜒 → ( 𝜏𝜂 ) ) )
5 1 a1d ( 𝜑 → ( 𝜃𝜓 ) )
6 2 a1d ( 𝜑 → ( 𝜃𝜒 ) )
7 5 6 3 4 ee222 ( 𝜑 → ( 𝜃𝜂 ) )