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Theorem ee33 39507
Description: Non-virtual deduction form of e33 39730. (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  39527  onfrALTlem2  39532  ee33an  39732  ee03  39737  ee30  39741  ee31  39748  ee32  39755  trintALT  39877
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