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Theorem frege34 37151
 Description: If as a conseqence of the occurence of the circumstance 𝜑, when the obstacle 𝜓 is removed, 𝜒 takes place, then from the circumstance that 𝜒 does not take place while 𝜑 occurs the occurence of the obstacle 𝜓 can be inferred. Closed form of con1d 138. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege34 ((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓)))

Proof of Theorem frege34
StepHypRef Expression
1 frege33 37150 . 2 ((¬ 𝜓𝜒) → (¬ 𝜒𝜓))
2 frege5 37114 . 2 (((¬ 𝜓𝜒) → (¬ 𝜒𝜓)) → ((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓))))
31, 2ax-mp 5 1 ((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4 This theorem was proved from axioms:  ax-mp 5  ax-frege1 37104  ax-frege2 37105  ax-frege28 37144  ax-frege31 37148 This theorem is referenced by:  frege35  37152  frege36  37153
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