Theorem ee33 45448

Description: Non-virtual deduction form of e33 45660. (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)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  truniALT  45468  onfrALTlem2  45473  ee33an  45662  ee03  45667  ee30  45671  ee31  45678  ee32  45685  trintALT  45807