Theorem frege67b

Description: Lemma for frege68b . Proposition 67 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege67b	$$⊢\left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \forall x\phi \right)\right)\to \left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \left[y/x\right]\phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ax-frege58b	$$⊢\forall x\phi \to \left[y/x\right]\phi$$
2		frege7	$$⊢\left(\forall x\phi \to \left[y/x\right]\phi \right)\to \left(\left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \forall x\phi \right)\right)\to \left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \left[y/x\right]\phi \right)\right)\right)$$
3	1 2	ax-mp	$$⊢\left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \forall x\phi \right)\right)\to \left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \left[y/x\right]\phi \right)\right)$$