Theorem frege64b

Description: Lemma for frege65b . Proposition 64 of Frege1879 p. 53. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege64b	$$⊢\left(\left[x/y\right]\phi \to \left[z/y\right]\psi \right)\to \left(\forall y\left(\psi \to \chi \right)\to \left(\left[x/y\right]\phi \to \left[z/y\right]\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege62b	$$⊢\left[z/y\right]\psi \to \left(\forall y\left(\psi \to \chi \right)\to \left[z/y\right]\chi \right)$$
2		frege18	$$⊢\left(\left[z/y\right]\psi \to \left(\forall y\left(\psi \to \chi \right)\to \left[z/y\right]\chi \right)\right)\to \left(\left(\left[x/y\right]\phi \to \left[z/y\right]\psi \right)\to \left(\forall y\left(\psi \to \chi \right)\to \left(\left[x/y\right]\phi \to \left[z/y\right]\chi \right)\right)\right)$$
3	1 2	ax-mp	$$⊢\left(\left[x/y\right]\phi \to \left[z/y\right]\psi \right)\to \left(\forall y\left(\psi \to \chi \right)\to \left(\left[x/y\right]\phi \to \left[z/y\right]\chi \right)\right)$$