Theorem frege59b

Description: A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of Frege1879 p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collectionFrom Frege to Goedel, this proof has the frege12 incorrectly referenced where frege30 is in the original. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege59b	$$⊢\left[y/x\right]\phi \to \left(\neg \left[y/x\right]\psi \to \neg \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege58bcor	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)$$
2		frege30	$$⊢\left(\forall x\left(\phi \to \psi \right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\right)\to \left(\left[y/x\right]\phi \to \left(\neg \left[y/x\right]\psi \to \neg \forall x\left(\phi \to \psi \right)\right)\right)$$
3	1 2	ax-mp	$$⊢\left[y/x\right]\phi \to \left(\neg \left[y/x\right]\psi \to \neg \forall x\left(\phi \to \psi \right)\right)$$