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Theorem pm4.14 600
 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 305 . . 3 ((𝜓𝜒) ↔ (¬ 𝜒 → ¬ 𝜓))
21imbi2i 325 . 2 ((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
3 impexp 461 . 2 (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))
4 impexp 461 . 2 (((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓) ↔ (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
52, 3, 43bitr4i 291 1 (((𝜑𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 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 196  df-an 385 This theorem is referenced by:  pm3.37  601  ndvdssub  14971
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