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Theorem imbi13 40839
Description: Join three logical equivalences to form equivalence of implications. imbi13 40839 is imbi13VD 41193 without virtual deductions and was automatically derived from imbi13VD 41193 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 349 . 2 ((𝜒𝜃) → ((𝜏𝜂) → ((𝜒𝜏) ↔ (𝜃𝜂))))
2 imbi12 349 . 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 208
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 209
This theorem is referenced by:  trsbc  40859  trsbcVD  41196
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