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Theorem ee233 44516
Description: Non-virtual deduction form of e233 44762. (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 44515 . . . . . . . 8 ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))))
97, 8mpbi 230 . . . . . . 7 (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))
10 pm2.43cbi 44515 . . . . . . 7 ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))
119, 10mpbi 230 . . . . . 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 44515 . . . 4 ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))))
1513, 14mpbi 230 . . 3 (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))
16 pm2.43cbi 44515 . . 3 ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))
1715, 16mpbi 230 . 2 (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))
18 pm2.43cbi 44515 . 2 ((𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))) ↔ (𝜑 → (𝜓 → (𝜃𝜁))))
1917, 18mpbi 230 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 207
This theorem is referenced by:  truniALT  44538  onfrALTlem2  44543  e233  44762
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