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Theorem pm4.14 599
Description: Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
Assertion
Ref Expression
pm4.14 (((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))

Proof of Theorem pm4.14
StepHypRef Expression
1 con34b 304 . . 3 ((𝜓𝜒) ↔ (¬ 𝜒 → ¬ 𝜓))
21imbi2i 324 . 2 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
3 impexp 460 . 2 (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))
4 impexp 460 . 2 (((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
52, 3, 43bitr4i 290 1 (((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382
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 195  df-an 384
This theorem is referenced by:  pm3.37  600  ndvdssub  14913
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