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Theorem frege46 37612
Description: If 𝜓 holds when 𝜑 occurs as well as when 𝜑 does not occur, then 𝜓 holds. If 𝜓 or 𝜑 occurs and if the occurences of 𝜑 has 𝜓 as a necessary consequence, then 𝜓 takes place. Identical to pm2.6 182. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege46 ((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓))

Proof of Theorem frege46
StepHypRef Expression
1 frege33 37598 . 2 ((¬ 𝜑𝜓) → (¬ 𝜓𝜑))
2 frege45 37611 . 2 (((¬ 𝜑𝜓) → (¬ 𝜓𝜑)) → ((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓)))
31, 2ax-mp 5 1 ((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-frege1 37552  ax-frege2 37553  ax-frege8 37571  ax-frege28 37592  ax-frege31 37596  ax-frege41 37607
This theorem is referenced by:  frege47  37613
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