Theorem ee233 45268

Description: Non-virtual deduction form of e233 45513. (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 36	. . . . . . . . . 10 ⊢ (𝜑 → (𝜓 → (𝜏 → (𝜂 → 𝜁))))
6	5	com3r 88	. . . . . . . . 9 ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))
7	2, 6	syl8 77	. . . . . . . 8 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))))
8		pm2.43cbi 45267	. . . . . . . 8 ⊢ ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))))
9	7, 8	mpbi 233	. . . . . . 7 ⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))
10		pm2.43cbi 45267	. . . . . . 7 ⊢ ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁)))))
11	9, 10	mpbi 233	. . . . . 6 ⊢ (𝜃 → (𝜑 → (𝜓 → (𝜂 → 𝜁))))
12	11	com14 97	. . . . 5 ⊢ (𝜂 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))
13	1, 12	syl8 77	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))))
14		pm2.43cbi 45267	. . . 4 ⊢ ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))))
15	13, 14	mpbi 233	. . 3 ⊢ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))
16		pm2.43cbi 45267	. . 3 ⊢ ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))))
17	15, 16	mpbi 233	. 2 ⊢ (𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁))))
18		pm2.43cbi 45267	. 2 ⊢ ((𝜃 → (𝜑 → (𝜓 → (𝜃 → 𝜁)))) ↔ (𝜑 → (𝜓 → (𝜃 → 𝜁))))
19	17, 18	mpbi 233	1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜁)))

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:  truniALT  45290  onfrALTlem2  45295  e233  45513