Theorem frege67c

Description: Lemma for frege68c . Proposition 67 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	$$⊢A\in B$$
Assertion	frege67c	$$⊢\left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \forall x\phi \right)\right)\to \left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	$$⊢A\in B$$
2	1	frege58c	$$⊢\forall x\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$
3		frege7	$$⊢\left(\forall x\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\to \left(\left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \forall x\phi \right)\right)\to \left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\right)\right)$$
4	2 3	ax-mp	$$⊢\left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \forall x\phi \right)\right)\to \left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\right)$$