Theorem frege33 41444

Description: If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 146. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege33	⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))

Proof of Theorem frege33

Step	Hyp	Ref	Expression
1		ax-frege28 41438	. 2 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑))
2		frege32 41443	. 2 ⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41398  ax-frege2 41399  ax-frege28 41438  ax-frege31 41442

This theorem is referenced by:  frege34  41445  frege46  41458