Theorem ee33 44948

Description: Non-virtual deduction form of e33 45160. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

h1::	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
h2::	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
h3::	⊢ (𝜃 → (𝜏 → 𝜂))
4:1,3:	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
5:4:	⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
6:2,5:	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
7:6:	⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
8:7:	⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
qed:8:	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Hypotheses

Ref	Expression
ee33.1	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
ee33.2	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
ee33.3	⊢ (𝜃 → (𝜏 → 𝜂))

Assertion

Ref	Expression
ee33	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Proof of Theorem ee33

Step	Hyp	Ref	Expression
1		ee33.1	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
2		ee33.2	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
3		ee33.3	. . 3 ⊢ (𝜃 → (𝜏 → 𝜂))
4	3	imim3i 64	. 2 ⊢ ((𝜒 → 𝜃) → ((𝜒 → 𝜏) → (𝜒 → 𝜂)))
5	1, 2, 4	syl6c 70	1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Syntax hints:   → wi 4

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

This theorem is referenced by:  truniALT  44968  onfrALTlem2  44973  ee33an  45162  ee03  45167  ee30  45171  ee31  45178  ee32  45185  trintALT  45307