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Theorem imbi12 334
Description: Closed form of imbi12i 338. 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 161 . . 3 (¬ ((𝜑𝜓) → ¬ (𝜒𝜃)) → (𝜑𝜓))
2 simprim 160 . . 3 (¬ ((𝜑𝜓) → ¬ (𝜒𝜃)) → (𝜒𝜃))
31, 2imbi12d 332 . 2 (¬ ((𝜑𝜓) → ¬ (𝜒𝜃)) → ((𝜑𝜒) ↔ (𝜓𝜃)))
43expi 159 1 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜑𝜒) ↔ (𝜓𝜃))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194
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 195
This theorem is referenced by:  imbi12i  338  bj-imbi12  31540  ifpbi12  36650  ifpbi13  36651  imbi13  37545  imbi13VD  37930  sbcssgVD  37939
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