Theorem frege62b

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara when the minor premise has a particular context. Proposition 62 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege62b	$$⊢\left[y/x\right]\phi \to \left(\forall x\left(\phi \to \psi \right)\to \left[y/x\right]\psi \right)$$

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		ax-frege8	$$⊢\left(\forall x\left(\phi \to \psi \right)\to \left(\left[y/x\right]\phi \to \left[y/x\right]\psi \right)\right)\to \left(\left[y/x\right]\phi \to \left(\forall x\left(\phi \to \psi \right)\to \left[y/x\right]\psi \right)\right)$$
3	1 2	ax-mp	$$⊢\left[y/x\right]\phi \to \left(\forall x\left(\phi \to \psi \right)\to \left[y/x\right]\psi \right)$$