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Theorem ee233 39220
Description: Non-virtual deduction form of e233 39486. (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 39219 . . . . . . . 8 ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))))
97, 8mpbi 221 . . . . . . 7 (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))
10 pm2.43cbi 39219 . . . . . . 7 ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))
119, 10mpbi 221 . . . . . 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 39219 . . . 4 ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))))
1513, 14mpbi 221 . . 3 (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))
16 pm2.43cbi 39219 . . 3 ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))
1715, 16mpbi 221 . 2 (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))
18 pm2.43cbi 39219 . 2 ((𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))) ↔ (𝜑 → (𝜓 → (𝜃𝜁))))
1917, 18mpbi 221 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 198
This theorem is referenced by:  truniALT  39246  onfrALTlem2  39256  e233  39486
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