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Theorem ee1111 41588
Description: Non-virtual deduction form of e1111 41747. (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) h2:: ⊢ (𝜑 → 𝜒) h3:: ⊢ (𝜑 → 𝜃) h4:: ⊢ (𝜑 → 𝜏) h5:: ⊢ (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))) 6:1,5: ⊢ (𝜑 → (𝜒 → (𝜃 → (𝜏 → 𝜂)))) 7:6: ⊢ (𝜒 → (𝜑 → (𝜃 → (𝜏 → 𝜂)))) 8:2,7: ⊢ (𝜑 → (𝜑 → (𝜃 → (𝜏 → 𝜂)))) 9:8: ⊢ (𝜑 → (𝜃 → (𝜏 → 𝜂))) 10:9: ⊢ (𝜃 → (𝜑 → (𝜏 → 𝜂))) 11:3,10: ⊢ (𝜑 → (𝜑 → (𝜏 → 𝜂))) 12:11: ⊢ (𝜑 → (𝜏 → 𝜂)) 13:12: ⊢ (𝜏 → (𝜑 → 𝜂)) 14:4,13: ⊢ (𝜑 → (𝜑 → 𝜂)) qed:14: ⊢ (𝜑 → 𝜂)
Hypotheses
Ref Expression
ee1111.1 (𝜑𝜓)
ee1111.2 (𝜑𝜒)
ee1111.3 (𝜑𝜃)
ee1111.4 (𝜑𝜏)
ee1111.5 (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))
Assertion
Ref Expression
ee1111 (𝜑𝜂)

Proof of Theorem ee1111
StepHypRef Expression
1 ee1111.4 . 2 (𝜑𝜏)
2 ee1111.1 . . 3 (𝜑𝜓)
3 ee1111.2 . . 3 (𝜑𝜒)
4 ee1111.3 . . 3 (𝜑𝜃)
5 ee1111.5 . . 3 (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))
62, 3, 4, 5syl3c 66 . 2 (𝜑 → (𝜏𝜂))
71, 6mpd 15 1 (𝜑𝜂)
 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:  e1111  41747
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