Theorem frege36 37153

Description: The case in which 𝜓 is denied, ¬ 𝜑 is affirmed, and 𝜑 is affirmed does not occur. If 𝜑 occurs, then (at least) one of the two, 𝜑 or 𝜓, takes place (no matter what 𝜓 might be). Identical to pm2.24 120. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege36	⊢ (𝜑 → (¬ 𝜑 → 𝜓))

Proof of Theorem frege36

Step	Hyp	Ref	Expression
1		ax-frege1 37104	. 2 ⊢ (𝜑 → (¬ 𝜓 → 𝜑))
2		frege34 37151	. 2 ⊢ ((𝜑 → (¬ 𝜓 → 𝜑)) → (𝜑 → (¬ 𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 37104  ax-frege2 37105  ax-frege28 37144  ax-frege31 37148

This theorem is referenced by:  frege37  37154  frege38  37155  frege83  37260