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Theorem ee33VD 39606
Description: Non-virtual deduction form of e33 39455. 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 39222 is ee33VD 39606 without virtual deductions and was automatically derived from ee33VD 39606.
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 39219 . . . . 5 ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))) ↔ (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
87biimpi 207 . . . 4 ((𝜑 → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))) → (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))))
96, 8e0a 39493 . . 3 (𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
10 pm2.43cbi 39219 . . . 4 ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))) ↔ (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
1110biimpi 207 . . 3 ((𝜓 → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))) → (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))))
129, 11e0a 39493 . 2 (𝜒 → (𝜑 → (𝜓 → (𝜒𝜂))))
13 pm2.43cbi 39219 . . 3 ((𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))) ↔ (𝜑 → (𝜓 → (𝜒𝜂))))
1413biimpi 207 . 2 ((𝜒 → (𝜑 → (𝜓 → (𝜒𝜂)))) → (𝜑 → (𝜓 → (𝜒𝜂))))
1512, 14e0a 39493 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 198
This theorem is referenced by: (None)
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