Theorem frege66b

Description: Swap antecedents of frege65b . Proposition 66 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege66b	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\chi \to \phi \right)\to \left(\left[y/x\right]\chi \to \left[y/x\right]\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege65b	$$⊢\forall x\left(\chi \to \phi \right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\left[y/x\right]\chi \to \left[y/x\right]\psi \right)\right)$$
2		ax-frege8	$$⊢\left(\forall x\left(\chi \to \phi \right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\left[y/x\right]\chi \to \left[y/x\right]\psi \right)\right)\right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\chi \to \phi \right)\to \left(\left[y/x\right]\chi \to \left[y/x\right]\psi \right)\right)\right)$$
3	1 2	ax-mp	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\chi \to \phi \right)\to \left(\left[y/x\right]\chi \to \left[y/x\right]\psi \right)\right)$$