Theorem frege64c

Description: Lemma for frege65c . Proposition 64 of Frege1879 p. 53. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	$$⊢A\in B$$
Assertion	frege64c	$$⊢\left(\underset{\dot{}}{[}C/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}C/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	$$⊢A\in B$$
2	1	frege62c	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \left(\forall x\left(\psi \to \chi \right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$
3		frege18	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \left(\forall x\left(\psi \to \chi \right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\to \left(\left(\underset{\dot{}}{[}C/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}C/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
4	2 3	ax-mp	$$⊢\left(\underset{\dot{}}{[}C/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}C/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$