Theorem frege68c

Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	$$⊢A\in B$$
Assertion	frege68c	$$⊢\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	$$⊢A\in B$$
2		frege57aid	$$⊢\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \forall x\phi \right)$$
3	1	frege67c	$$⊢\left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \forall x\phi \right)\right)\to \left(\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\right)$$
4	2 3	ax-mp	$$⊢\left(\forall x\phi \leftrightarrow \psi \right)\to \left(\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$