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Theorem ee233 41633
Description: Non-virtual deduction form of e233 41879. (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 → (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 41632 . . . . . . . 8 ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))))
97, 8mpbi 233 . . . . . . 7 (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))
10 pm2.43cbi 41632 . . . . . . 7 ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜂𝜁)))))
119, 10mpbi 233 . . . . . 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 41632 . . . 4 ((𝜑 → (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))) ↔ (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))))
1513, 14mpbi 233 . . 3 (𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))
16 pm2.43cbi 41632 . . 3 ((𝜓 → (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))) ↔ (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))))
1715, 16mpbi 233 . 2 (𝜃 → (𝜑 → (𝜓 → (𝜃𝜁))))
18 pm2.43cbi 41632 . 2 ((𝜃 → (𝜑 → (𝜓 → (𝜃𝜁)))) ↔ (𝜑 → (𝜓 → (𝜃𝜁))))
1917, 18mpbi 233 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 210 This theorem is referenced by:  truniALT  41655  onfrALTlem2  41660  e233  41879
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