Metamath Proof Explorer


Theorem ee13an

Description: e13an without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ee13an.1 ( 𝜑𝜓 )
ee13an.2 ( 𝜑 → ( 𝜒 → ( 𝜃𝜏 ) ) )
ee13an.3 ( ( 𝜓𝜏 ) → 𝜂 )
Assertion ee13an ( 𝜑 → ( 𝜒 → ( 𝜃𝜂 ) ) )

Proof

Step Hyp Ref Expression
1 ee13an.1 ( 𝜑𝜓 )
2 ee13an.2 ( 𝜑 → ( 𝜒 → ( 𝜃𝜏 ) ) )
3 ee13an.3 ( ( 𝜓𝜏 ) → 𝜂 )
4 3 ex ( 𝜓 → ( 𝜏𝜂 ) )
5 1 2 4 ee13 ( 𝜑 → ( 𝜒 → ( 𝜃𝜂 ) ) )