Theorem ee31an 41081

Description: e31an 41080 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ee31an.1	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
ee31an.2	⊢ (𝜑 → 𝜏)
ee31an.3	⊢ ((𝜃 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
ee31an	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Proof of Theorem ee31an

Step	Hyp	Ref	Expression
1		ee31an.1	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
2		ee31an.2	. . . 4 ⊢ (𝜑 → 𝜏)
3	2	a1d 25	. . 3 ⊢ (𝜑 → (𝜒 → 𝜏))
4	3	a1d 25	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
5		ee31an.3	. 2 ⊢ ((𝜃 ∧ 𝜏) → 𝜂)
6	1, 4, 5	ee33an 41063	1 ⊢ (𝜑 → (𝜓 → (𝜒 → 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: (None)