Theorem ee233 42139

Description: Non-virtual deduction form of e233 42385. (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))
h2::	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
h3::	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))
h4::	⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))
5:1,4:	⊢ (𝜑 → (𝜓 → (𝜏 → (𝜂 → 𝜁))) )
6:5:	⊢ (𝜏 → (𝜑 → (𝜓 → (𝜂 → 𝜁))) )
7:2,6:	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))))
8:7:	⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))
9:8:	⊢ (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))) )
10:9:	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜂 → 𝜁))) )
11:10:	⊢ (𝜂 → (𝜑 → (𝜓 → (𝜃 → 𝜁))) )
12:3,11:	⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))))
13:12:	⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))
14:13:	⊢ (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))) )
qed:14:	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁)))

Hypotheses

Ref	Expression
ee233.1	⊢ (𝜑 → (𝜓 → 𝜒))
ee233.2	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
ee233.3	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))
ee233.4	⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))

Assertion

Ref	Expression
ee233	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁)))

Proof of Theorem ee233

Step	Hyp	Ref	Expression
1		ee233.3	. . . . 5 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))
2		ee233.2	. . . . . . . . 9 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏)))
3		ee233.1	. . . . . . . . . . 11 ⊢ (𝜑 → (𝜓 → 𝜒))
4		ee233.4	. . . . . . . . . . 11 ⊢ (𝜒 → (𝜏 → (𝜂 → 𝜁)))
5	3, 4	syl6 35	. . . . . . . . . 10 ⊢ (𝜑 → (𝜓 → (𝜏 → (𝜂 → 𝜁))))
6	5	com3r 87	. . . . . . . . 9 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))
7	2, 6	syl8 76	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))))
8		pm2.43cbi 42138	. . . . . . . 8 ⊢ ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))))
9	7, 8	mpbi 229	. . . . . . 7 ⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))
10		pm2.43cbi 42138	. . . . . . 7 ⊢ ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))
11	9, 10	mpbi 229	. . . . . 6 ⊢ (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))
12	11	com14 96	. . . . 5 ⊢ (𝜂 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))
13	1, 12	syl8 76	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))))
14		pm2.43cbi 42138	. . . 4 ⊢ ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))))
15	13, 14	mpbi 229	. . 3 ⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))
16		pm2.43cbi 42138	. . 3 ⊢ ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))
17	15, 16	mpbi 229	. 2 ⊢ (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))
18		pm2.43cbi 42138	. 2 ⊢ ((𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))) ↔ (𝜑 → (𝜓 → (𝜃 → 𝜁))))
19	17, 18	mpbi 229	1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁)))

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:  truniALT  42161  onfrALTlem2  42166  e233  42385