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Theorem ee233 38551
Description: Non-virtual deduction form of e233 38818. (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 → (𝜓𝜒))
h2:: (𝜑 → (𝜓 → (𝜃𝜏)))
h3:: (𝜑 → (𝜓 → (𝜃𝜂)))
h4:: (𝜒 → (𝜏 → (𝜂𝜁)))
5:1,4: (𝜑 → (𝜓 → (𝜏 → (𝜂𝜁))) )
6:5: (𝜏 → (𝜑 → (𝜓 → (𝜂𝜁))) )
7:2,6: (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))))
8:7: (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂 𝜁)))))
9:8: (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))) )
10:9: (𝜑 → (𝜓 → (𝜃 → (𝜂𝜁))) )
11:10: (𝜂 → (𝜑 → (𝜓 → (𝜃𝜁))) )
12:3,11: (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))))
13:12: (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃 𝜁)))))
14:13: (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))) )
qed:14: (𝜑 → (𝜓 → (𝜃𝜁)))
Hypotheses
Ref Expression
ee233.1 (𝜑 → (𝜓𝜒))
ee233.2 (𝜑 → (𝜓 → (𝜃𝜏)))
ee233.3 (𝜑 → (𝜓 → (𝜃𝜂)))
ee233.4 (𝜒 → (𝜏 → (𝜂𝜁)))
Assertion
Ref Expression
ee233 (𝜑 → (𝜓 → (𝜃𝜁)))

Proof of Theorem ee233
StepHypRef Expression
1 ee233.3 . . . . 5 (𝜑 → (𝜓 → (𝜃𝜂)))
2 ee233.2 . . . . . . . . 9 (𝜑 → (𝜓 → (𝜃𝜏)))
3 ee233.1 . . . . . . . . . . 11 (𝜑 → (𝜓𝜒))
4 ee233.4 . . . . . . . . . . 11 (𝜒 → (𝜏 → (𝜂𝜁)))
53, 4syl6 35 . . . . . . . . . 10 (𝜑 → (𝜓 → (𝜏 → (𝜂𝜁))))
65com3r 87 . . . . . . . . 9 (𝜏 → (𝜑 → (𝜓 → (𝜂𝜁))))
72, 6syl8 76 . . . . . . . 8 (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))))
8 pm2.43cbi 38550 . . . . . . . 8 ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))))
97, 8mpbi 220 . . . . . . 7 (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))
10 pm2.43cbi 38550 . . . . . . 7 ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))
119, 10mpbi 220 . . . . . 6 (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))
1211com14 96 . . . . 5 (𝜂 → (𝜑 → (𝜓 → (𝜃𝜁))))
131, 12syl8 76 . . . 4 (𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))))
14 pm2.43cbi 38550 . . . 4 ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))))
1513, 14mpbi 220 . . 3 (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))
16 pm2.43cbi 38550 . . 3 ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))
1715, 16mpbi 220 . 2 (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))
18 pm2.43cbi 38550 . 2 ((𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))) ↔ (𝜑 → (𝜓 → (𝜃𝜁))))
1917, 18mpbi 220 1 (𝜑 → (𝜓 → (𝜃𝜁)))
Colors of variables: wff setvar class
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 197
This theorem is referenced by:  truniALT  38577  onfrALTlem2  38587  e233  38818
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