Theorem frege44 45038

Description: Similar to a commuted pm2.62 913. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege44	⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜑))

Proof of Theorem frege44

Step	Hyp	Ref	Expression
1		frege43 45037	. 2 ⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
2		frege21 45017	. 2 ⊢ (((¬ 𝜑 → 𝜑) → 𝜑) → ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜑))

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:  frege45  45039