Theorem frege34 44591

Description: If as a consequence 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 146. 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 44590	. 2 ⊢ ((¬ 𝜓 → 𝜒) → (¬ 𝜒 → 𝜓))
2		frege5 44554	. 2 ⊢ (((¬ 𝜓 → 𝜒) → (¬ 𝜒 → 𝜓)) → ((𝜑 → (¬ 𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 44544  ax-frege2 44545  ax-frege28 44584  ax-frege31 44588

This theorem is used by:  frege35  44592  frege36  44593