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Theorem pm2.43bgbi 41310
Description: Logical equivalence of a 2-left-nested implication and a 1-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 → (𝜑 → 𝜒))) → (𝜑 → (𝜑 → (𝜓 → 𝜒)))) 2:: ⊢ ((𝜑 → (𝜑 → (𝜓 → 𝜒))) → (𝜑 → (𝜓 → 𝜒))) 3:1,2: ⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜑 → (𝜓 → 𝜒))) 4:: ⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒))) 5:3,4: ⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) → (𝜓 → (𝜑 → 𝜒))) 6:: ⊢ ((𝜓 → (𝜑 → 𝜒)) → (𝜑 → (𝜓 → (𝜑 → 𝜒)))) qed:5,6: ⊢ ((𝜑 → (𝜓 → (𝜑 → 𝜒))) ↔ (𝜓 → (𝜑 → 𝜒)))
Assertion
Ref Expression
pm2.43bgbi ((𝜑 → (𝜓 → (𝜑𝜒))) ↔ (𝜓 → (𝜑𝜒)))

Proof of Theorem pm2.43bgbi
StepHypRef Expression
1 mercolem6 1744 . 2 ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))
2 ax-1 6 . 2 ((𝜓 → (𝜑𝜒)) → (𝜑 → (𝜓 → (𝜑𝜒))))
31, 2impbii 212 1 ((𝜑 → (𝜓 → (𝜑𝜒))) ↔ (𝜓 → (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209 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 210  df-tru 1541  df-fal 1551 This theorem is referenced by: (None)
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