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Theorem ee33 44519
Description: Non-virtual deduction form of e33 44732. (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 64 . 2 ((𝜒𝜃) → ((𝜒𝜏) → (𝜒𝜂)))
51, 2, 4syl6c 70 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  44539  onfrALTlem2  44544  ee33an  44734  ee03  44739  ee30  44743  ee31  44750  ee32  44757  trintALT  44879
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