Theorem frege61c

Description: Lemma for frege65c . Proposition 61 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	$$⊢A\in B$$
Assertion	frege61c	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \psi \right)\to \left(\forall x\phi \to \psi \right)$$

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		frege9	$$⊢\left(\forall x\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\to \left(\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \psi \right)\to \left(\forall x\phi \to \psi \right)\right)$$
4	2 3	ax-mp	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \psi \right)\to \left(\forall x\phi \to \psi \right)$$