Theorem ee33VD 42388

Description: Non-virtual deduction form of e33 42243. 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 42030 is ee33VD 42388 without virtual deductions and was automatically derived from ee33VD 42388.

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 42027	. . . . 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 42281	. . 3 ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
10		pm2.43cbi 42027	. . . 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 42281	. 2 ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
13		pm2.43cbi 42027	. . 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 42281	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)