Theorem imp5a 443

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

Hypothesis

Ref	Expression
imp5.1	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))))

Assertion

Ref	Expression
imp5a	⊢ (𝜑 → (𝜓 → (𝜒 → ((𝜃 ∧ 𝜏) → 𝜂))))

Proof of Theorem imp5a

Step	Hyp	Ref	Expression
1		imp5.1	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))))
2	1	imp5d 442	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → ((𝜃 ∧ 𝜏) → 𝜂))
3	2	exp31 422	1 ⊢ (𝜑 → (𝜓 → (𝜒 → ((𝜃 ∧ 𝜏) → 𝜂))))

Syntax hints:   → wi 4   ∧ wa 398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 399

This theorem is referenced by:  prtlem17  36006  tendospcanN  38153