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Theorem imdistan 568
Description: Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
Assertion
Ref Expression
imdistan ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))

Proof of Theorem imdistan
StepHypRef Expression
1 pm5.42 544 . 2 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (𝜓 → (𝜑𝜒))))
2 impexp 451 . 2 (((𝜑𝜓) → (𝜑𝜒)) ↔ (𝜑 → (𝜓 → (𝜑𝜒))))
31, 2bitr4i 279 1 ((𝜑 → (𝜓𝜒)) ↔ ((𝜑𝜓) → (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396
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 208  df-an 397
This theorem is referenced by:  imdistand  571  rmoim  3728  ss2rab  4044  marypha2lem3  8889  inxpss3  35452
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