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Theorem ee33 37748
Description: Non-virtual deduction form of e33 37982. (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 → (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)))
Hypotheses
Ref Expression
ee33.1 (𝜑 → (𝜓 → (𝜒𝜃)))
ee33.2 (𝜑 → (𝜓 → (𝜒𝜏)))
ee33.3 (𝜃 → (𝜏𝜂))
Assertion
Ref Expression
ee33 (𝜑 → (𝜓 → (𝜒𝜂)))

Proof of Theorem ee33
StepHypRef Expression
1 ee33.1 . 2 (𝜑 → (𝜓 → (𝜒𝜃)))
2 ee33.2 . 2 (𝜑 → (𝜓 → (𝜒𝜏)))
3 ee33.3 . . 3 (𝜃 → (𝜏𝜂))
43imim3i 62 . 2 ((𝜒𝜃) → ((𝜒𝜏) → (𝜒𝜂)))
51, 2, 4syl6c 68 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
This theorem is referenced by:  truniALT  37772  onfrALTlem2  37782  ee33an  37984  ee03  37989  ee30  37993  ee31  38000  ee32  38007  trintALT  38139
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