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Theorem pm2.43cbi 41172
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 → (𝜑𝜃))) ) → (𝜑 → (𝜓 → (𝜑 → (𝜒𝜃)))))
2:: ((𝜑 → (𝜓 → (𝜑 → (𝜒𝜃))) ) → (𝜓 → (𝜑 → (𝜒𝜃))))
3:1,2: ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃))) ) → (𝜓 → (𝜑 → (𝜒𝜃))))
4:: ((𝜓 → (𝜑 → (𝜒𝜃))) → (𝜓 → (𝜒 → (𝜑𝜃))))
5:3,4: ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃))) ) → (𝜓 → (𝜒 → (𝜑𝜃))))
6:: ((𝜓 → (𝜒 → (𝜑𝜃))) → (𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))))
qed:5,6: ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃))) ) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))
Assertion
Ref Expression
pm2.43cbi ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))

Proof of Theorem pm2.43cbi
StepHypRef Expression
1 pm2.24 124 . . . 4 (𝜑 → (¬ 𝜑 → (𝜓 → (𝜒𝜃))))
21com4l 92 . . 3 𝜑 → (𝜓 → (𝜒 → (𝜑𝜃))))
3 id 22 . . 3 ((𝜓 → (𝜒 → (𝜑𝜃))) → (𝜓 → (𝜒 → (𝜑𝜃))))
42, 3ja 189 . 2 ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))) → (𝜓 → (𝜒 → (𝜑𝜃))))
5 ax-1 6 . 2 ((𝜓 → (𝜒 → (𝜑𝜃))) → (𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))))
64, 5impbii 212 1 ((𝜑 → (𝜓 → (𝜒 → (𝜑𝜃)))) ↔ (𝜓 → (𝜒 → (𝜑𝜃))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  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
This theorem is referenced by:  ee233  41173  ee33VD  41533
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