Theorem ee33 45102

Description: Non-virtual deduction form of e33 45314. (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  45122  onfrALTlem2  45127  ee33an  45316  ee03  45321  ee30  45325  ee31  45332  ee32  45339  trintALT  45461