Theorem frege11 41422

Description: Elimination of a nested antecedent as a partial converse of ja 186. If the proposition that 𝜓 takes place or 𝜑 does not is a sufficient condition for 𝜒, then 𝜓 by itself is a sufficient condition for 𝜒. Identical to jarr 106. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege11	⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))

Proof of Theorem frege11

Step	Hyp	Ref	Expression
1		ax-frege1 41398	. 2 ⊢ (𝜓 → (𝜑 → 𝜓))
2		frege9 41420	. 2 ⊢ ((𝜓 → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)))
3	1, 2	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41398  ax-frege2 41399  ax-frege8 41417

This theorem is referenced by:  frege112  41583