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Theorem imimorb 920
Description: Simplify an implication between implications. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
Assertion
Ref Expression
imimorb (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))

Proof of Theorem imimorb
StepHypRef Expression
1 bi2.04 376 . 2 (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒)))
2 dfor2 427 . . 3 ((𝜓𝜒) ↔ ((𝜓𝜒) → 𝜒))
32imbi2i 326 . 2 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → ((𝜓𝜒) → 𝜒)))
41, 3bitr4i 267 1 (((𝜓𝜒) → (𝜑𝜒)) ↔ (𝜑 → (𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383
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  df-or 385
This theorem is referenced by: (None)
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