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Theorem ee33 38547
Description: Non-virtual deduction form of e33 38781. (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 → (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)))
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  38571  onfrALTlem2  38581  ee33an  38783  ee03  38788  ee30  38792  ee31  38799  ee32  38806  trintALT  38937
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