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Theorem ee33VD 40632
Description: Non-virtual deduction form of e33 40487. 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. ee33 40274 is ee33VD 40632 without virtual deductions and was automatically derived from ee33VD 40632.
 h1:: ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃))) h2:: ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏))) h3:: ⊢ (𝜃 → (𝜏 → 𝜂)) 4:1,3: ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜏 → 𝜂)))) 5:4: ⊢ (𝜏 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) 6:2,5: ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))))) 7:6: ⊢ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂))))) 8:7: ⊢ (𝜒 → (𝜑 → (𝜓 → (𝜒 → 𝜂)))) qed:8: ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))
(Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ee33VD.1 (𝜑 → (𝜓 → (𝜒𝜃)))
ee33VD.2 (𝜑 → (𝜓 → (𝜒𝜏)))
ee33VD.3 (𝜃 → (𝜏𝜂))
Assertion
Ref Expression
ee33VD (𝜑 → (𝜓 → (𝜒𝜂)))

Proof of Theorem ee33VD
StepHypRef Expression
1 ee33VD.2 . . . . 5 (𝜑 → (𝜓 → (𝜒𝜏)))
2 ee33VD.1 . . . . . . 7 (𝜑 → (𝜓 → (𝜒𝜃)))
3 ee33VD.3 . . . . . . 7 (𝜃 → (𝜏𝜂))
42, 3syl8 76 . . . . . 6 (𝜑 → (𝜓 → (𝜒 → (𝜏𝜂))))
54com4r 94 . . . . 5 (𝜏 → (𝜑 → (𝜓 → (𝜒𝜂))))
61, 5syl8 76 . . . 4 (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
7 pm2.43cbi 40271 . . . . 5 ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))) ↔ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
87biimpi 208 . . . 4 ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))) → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
96, 8e0a 40525 . . 3 (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
10 pm2.43cbi 40271 . . . 4 ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))) ↔ (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
1110biimpi 208 . . 3 ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))) → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
129, 11e0a 40525 . 2 (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))
13 pm2.43cbi 40271 . . 3 ((𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))) ↔ (𝜑 → (𝜓 → (𝜒𝜂))))
1413biimpi 208 . 2 ((𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))) → (𝜑 → (𝜓 → (𝜒𝜂))))
1512, 14e0a 40525 1 (𝜑 → (𝜓 → (𝜒𝜂)))
 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 199 This theorem is referenced by: (None)
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