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Theorem imbi12 335
 Description: Closed form of imbi12i 339. Was automatically derived from its "Virtual Deduction" version and Metamath's "minimize" command. (Contributed by Alan Sare, 18-Mar-2012.)
Assertion
Ref Expression
imbi12 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜑𝜒) ↔ (𝜓𝜃))))

Proof of Theorem imbi12
StepHypRef Expression
1 simplim 164 . . 3 (¬ ((𝜑𝜓) → ¬ (𝜒𝜃)) → (𝜑𝜓))
2 simprim 163 . . 3 (¬ ((𝜑𝜓) → ¬ (𝜒𝜃)) → (𝜒𝜃))
31, 2imbi12d 333 . 2 (¬ ((𝜑𝜓) → ¬ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))
43expi 162 1 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜑𝜒) ↔ (𝜓𝜃))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196 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 197 This theorem is referenced by:  imbi12i  339  bj-imbi12  32904  ifpbi12  38359  ifpbi13  38360  imbi13  39251  imbi13VD  39632  sbcssgVD  39641
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