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Theorem ee33VD 38137
Description: Non-virtual deduction form of e33 37982. 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 37748 is ee33VD 38137 without virtual deductions and was automatically derived from ee33VD 38137.
 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 74 . . . . . 6 (𝜑 → (𝜓 → (𝜒 → (𝜏𝜂))))
54com4r 92 . . . . 5 (𝜏 → (𝜑 → (𝜓 → (𝜒𝜂))))
61, 5syl8 74 . . . 4 (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
7 pm2.43cbi 37745 . . . . 5 ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))) ↔ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
87biimpi 205 . . . 4 ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))) → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
96, 8e0a 38020 . . 3 (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
10 pm2.43cbi 37745 . . . 4 ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))) ↔ (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
1110biimpi 205 . . 3 ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))) → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
129, 11e0a 38020 . 2 (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))
13 pm2.43cbi 37745 . . 3 ((𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))) ↔ (𝜑 → (𝜓 → (𝜒𝜂))))
1413biimpi 205 . 2 ((𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))) → (𝜑 → (𝜓 → (𝜒𝜂))))
1512, 14e0a 38020 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 196 This theorem is referenced by: (None)
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