 Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm2.43cbi Structured version   Visualization version   GIF version

Theorem pm2.43cbi 37745
Description: Logical equivalence of a 3-left-nested implication and a 2-left-nested implicated when two antecedents of the former implication are identical. (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.
 1:: ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜑 → (𝜓 → (𝜑 → (𝜒 → 𝜃))))) 2:: ⊢ ((𝜑 → (𝜓 → (𝜑 → (𝜒 → 𝜃))) ) → (𝜓 → (𝜑 → (𝜒 → 𝜃)))) 3:1,2: ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜓 → (𝜑 → (𝜒 → 𝜃)))) 4:: ⊢ ((𝜓 → (𝜑 → (𝜒 → 𝜃))) → (𝜓 → (𝜒 → (𝜑 → 𝜃)))) 5:3,4: ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) → (𝜓 → (𝜒 → (𝜑 → 𝜃)))) 6:: ⊢ ((𝜓 → (𝜒 → (𝜑 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))))) qed:5,6: ⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜑 → 𝜃))) ) ↔ (𝜓 → (𝜒 → (𝜑 → 𝜃))))
Assertion
Ref Expression
pm2.43cbi ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))

Proof of Theorem pm2.43cbi
StepHypRef Expression
1 pm2.24 120 . . . 4 (𝜑 → (¬ 𝜑 → (𝜓 → (𝜒𝜃))))
21com4l 90 . . 3 𝜑 → (𝜓 → (𝜒 → (𝜑𝜃))))
3 id 22 . . 3 ((𝜓 → (𝜒 → (𝜑𝜃))) → (𝜓 → (𝜒 → (𝜑𝜃))))
42, 3ja 172 . 2 ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))) → (𝜓 → (𝜒 → (𝜑𝜃))))
5 ax-1 6 . 2 ((𝜓 → (𝜒 → (𝜑𝜃))) → (𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))))
64, 5impbii 198 1 ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 196 This theorem is referenced by:  ee233  37746  ee33VD  38137
 Copyright terms: Public domain W3C validator