Theorem ee33 45156

Description: Non-virtual deduction form of e33 45368. (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 65	. 2 ⊢ ((𝜒 → 𝜃) → ((𝜒 → 𝜏) → (𝜒 → 𝜂)))
5	1, 2, 4	syl6c 71	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  45176  onfrALTlem2  45181  ee33an  45370  ee03  45375  ee30  45379  ee31  45386  ee32  45393  trintALT  45515