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Theorem imbi13 39559
 Description: Join three logical equivalences to form equivalence of implications. imbi13 39559 is imbi13VD 39923 without virtual deductions and was automatically derived from imbi13VD 39923 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
imbi13 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))

Proof of Theorem imbi13
StepHypRef Expression
1 imbi12 338 . 2 ((𝜒𝜃) → ((𝜏𝜂) → ((𝜒𝜏) ↔ (𝜃𝜂))))
2 imbi12 338 . 2 ((𝜑𝜓) → (((𝜒𝜏) ↔ (𝜃𝜂)) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂)))))
31, 2syl9r 78 1 ((𝜑𝜓) → ((𝜒𝜃) → ((𝜏𝜂) → ((𝜑 → (𝜒𝜏)) ↔ (𝜓 → (𝜃𝜂))))))
 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:  trsbc  39579  trsbcVD  39926
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