Metamath Proof Explorer


Theorem imp5a

Description: An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009) (Proof shortened by Wolf Lammen, 2-Aug-2022)

Ref Expression
Hypothesis imp5.1 ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃 → ( 𝜏𝜂 ) ) ) ) )
Assertion imp5a ( 𝜑 → ( 𝜓 → ( 𝜒 → ( ( 𝜃𝜏 ) → 𝜂 ) ) ) )

Proof

Step Hyp Ref Expression
1 imp5.1 ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃 → ( 𝜏𝜂 ) ) ) ) )
2 1 imp5d ( ( ( 𝜑𝜓 ) ∧ 𝜒 ) → ( ( 𝜃𝜏 ) → 𝜂 ) )
3 2 exp31 ( 𝜑 → ( 𝜓 → ( 𝜒 → ( ( 𝜃𝜏 ) → 𝜂 ) ) ) )