Theorem frege34 41334

Description: If as a conseqence of the occurrence 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 occurrence of the obstacle 𝜓 can be inferred. Closed form of con1d 145. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege34	⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → 𝜓)))

Proof of Theorem frege34

Step	Hyp	Ref	Expression
1		frege33 41333	. 2 ⊢ ((¬ 𝜓 → 𝜒) → (¬ 𝜒 → 𝜓))
2		frege5 41297	. 2 ⊢ (((¬ 𝜓 → 𝜒) → (¬ 𝜒 → 𝜓)) → ((𝜑 → (¬ 𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41287  ax-frege2 41288  ax-frege28 41327  ax-frege31 41331

This theorem is referenced by:  frege35  41335  frege36  41336