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Theorem ee33VD 38935
Description: Non-virtual deduction form of e33 38781. 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 38547 is ee33VD 38935 without virtual deductions and was automatically derived from ee33VD 38935.
h1:: (𝜑 → (𝜓 → (𝜒𝜃)))
h2:: (𝜑 → (𝜓 → (𝜒𝜏)))
h3:: (𝜃 → (𝜏𝜂))
4:1,3: (𝜑 → (𝜓 → (𝜒 → (𝜏𝜂))))
5:4: (𝜏 → (𝜑 → (𝜓 → (𝜒𝜂))))
6:2,5: (𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
7:6: (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒 𝜂)))))
8:7: (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))
qed:8: (𝜑 → (𝜓 → (𝜒𝜂)))
(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 38544 . . . . 5 ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))) ↔ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
87biimpi 206 . . . 4 ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))) → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
96, 8e0a 38819 . . 3 (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
10 pm2.43cbi 38544 . . . 4 ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))) ↔ (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
1110biimpi 206 . . 3 ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))) → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
129, 11e0a 38819 . 2 (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))
13 pm2.43cbi 38544 . . 3 ((𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))) ↔ (𝜑 → (𝜓 → (𝜒𝜂))))
1413biimpi 206 . 2 ((𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))) → (𝜑 → (𝜓 → (𝜒𝜂))))
1512, 14e0a 38819 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 197
This theorem is referenced by: (None)
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