Theorem ee33VD 45702

Description: Non-virtual deduction form of e33 45557. 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 45345 is ee33VD 45702 without virtual deductions and was automatically derived from ee33VD 45702.

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 77	. . . . . 6 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂))))
5	4	com4r 95	. . . . 5 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
6	1, 5	syl8 77	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
7		pm2.43cbi 45342	. . . . 5 ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) ↔ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
8	7	biimpi 219	. . . 4 ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))))
9	6, 8	e0a 45595	. . 3 ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
10		pm2.43cbi 45342	. . . 4 ⊢ ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) ↔ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
11	10	biimpi 219	. . 3 ⊢ ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))
12	9, 11	e0a 45595	. 2 ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
13		pm2.43cbi 45342	. . 3 ⊢ ((𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) ↔ (𝜑 → (𝜓 → (𝜒 → 𝜂))))
14	13	biimpi 219	. 2 ⊢ ((𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) → (𝜑 → (𝜓 → (𝜒 → 𝜂))))
15	12, 14	e0a 45595	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  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by: (None)