Theorem frege46 41458

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 occurrences of 𝜑 has 𝜓 as a necessary consequence, then 𝜓 takes place. Identical to pm2.6 190. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege46	⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))

Proof of Theorem frege46

Step	Hyp	Ref	Expression
1		frege33 41444	. 2 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))
2		frege45 41457	. 2 ⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41398  ax-frege2 41399  ax-frege8 41417  ax-frege28 41438  ax-frege31 41442  ax-frege41 41453

This theorem is referenced by:  frege47  41459