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Theorem com3rgbi 40304
Description: 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. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃)))) 2:: ⊢ ((𝜑 → (𝜒 → (𝜓 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃)))) 3:1,2: ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃)))) 4:: ⊢ ((𝜒 → (𝜑 → (𝜓 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃)))) 5:: ⊢ ((𝜑 → (𝜒 → (𝜓 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → 𝜃)))) 6:4,5: ⊢ ((𝜒 → (𝜑 → (𝜓 → 𝜃))) → (𝜑 → (𝜓 → (𝜒 → 𝜃)))) qed:3,6: ⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) ↔ (𝜒 → (𝜑 → (𝜓 → 𝜃))))
Assertion
Ref Expression
com3rgbi ((𝜑 → (𝜓 → (𝜒𝜃))) ↔ (𝜒 → (𝜑 → (𝜓𝜃))))

Proof of Theorem com3rgbi
StepHypRef Expression
1 pm2.04 90 . . 3 ((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜓 → (𝜑 → (𝜒𝜃))))
21com24 95 . 2 ((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜒 → (𝜑 → (𝜓𝜃))))
3 pm2.04 90 . . 3 ((𝜒 → (𝜑 → (𝜓𝜃))) → (𝜑 → (𝜒 → (𝜓𝜃))))
43com34 91 . 2 ((𝜒 → (𝜑 → (𝜓𝜃))) → (𝜑 → (𝜓 → (𝜒𝜃))))
52, 4impbii 201 1 ((𝜑 → (𝜓 → (𝜒𝜃))) ↔ (𝜒 → (𝜑 → (𝜓𝜃))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198 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 199 This theorem is referenced by:  impexpdcom  40305
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