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Theorem impbidd 199
 Description: Deduce an equivalence from two implications. Double deduction associated with impbi 197 and impbii 198. Deduction associated with impbid 201. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Hypotheses
Ref Expression
impbidd.1 (𝜑 → (𝜓 → (𝜒𝜃)))
impbidd.2 (𝜑 → (𝜓 → (𝜃𝜒)))
Assertion
Ref Expression
impbidd (𝜑 → (𝜓 → (𝜒𝜃)))

Proof of Theorem impbidd
StepHypRef Expression
1 impbidd.1 . 2 (𝜑 → (𝜓 → (𝜒𝜃)))
2 impbidd.2 . 2 (𝜑 → (𝜓 → (𝜃𝜒)))
3 impbi 197 . 2 ((𝜒𝜃) → ((𝜃𝜒) → (𝜒𝜃)))
41, 2, 3syl6c 68 1 (𝜑 → (𝜓 → (𝜒𝜃)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195 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 196 This theorem is referenced by:  impbid21d  200  pm5.74  258  seglecgr12  31388  prtlem18  33180
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