Theorem ee33VD 43942

Description: Non-virtual deduction form of e33 43797. 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. ee33 43584 is ee33VD 43942 without virtual deductions and was automatically derived from ee33VD 43942.

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)))

(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ee33VD.1	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
ee33VD.2	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
ee33VD.3	⊢ (𝜃 → (𝜏 → 𝜂))

Assertion

Ref	Expression
ee33VD	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Proof of Theorem ee33VD

Step	Hyp	Ref	Expression
1		ee33VD.2	. . . . 5 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
2		ee33VD.1	. . . . . . 7 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3		ee33VD.3	. . . . . . 7 ⊢ (𝜃 → (𝜏 → 𝜂))
4	2, 3	syl8 76	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
5	4	com4r 94	. . . . 5 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
6	1, 5	syl8 76	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
7		pm2.43cbi 43581	. . . . 5 ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) ↔ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
8	7	biimpi 215	. . . 4 ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
9	6, 8	e0a 43835	. . 3 ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
10		pm2.43cbi 43581	. . . 4 ⊢ ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) ↔ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
11	10	biimpi 215	. . 3 ⊢ ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
12	9, 11	e0a 43835	. 2 ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
13		pm2.43cbi 43581	. . 3 ⊢ ((𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜂))))
14	13	biimpi 215	. 2 ⊢ ((𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
15	12, 14	e0a 43835	1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))

Syntax hints:   → wi 4

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 206

This theorem is referenced by: (None)