Theorem frege46 45040

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 pm2.6 193. 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 45026	. 2 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))
2		frege45 45039	. 2 ⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))

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

This proof depends on axioms:  ax-mp 5  ax-frege1 44980  ax-frege2 44981  ax-frege8 44999  ax-frege28 45020  ax-frege31 45024  ax-frege41 45035

This theorem is used by:  frege47  45041