Theorem frege66c

Description: Swap antecedents of frege65c . Proposition 66 of Frege1879 p. 54. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	$$⊢A\in B$$
Assertion	frege66c	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\chi \to \phi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	$$⊢A\in B$$
2	1	frege65c	$$⊢\forall x\left(\chi \to \phi \right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$
3		ax-frege8	$$⊢\left(\forall x\left(\chi \to \phi \right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)\right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\chi \to \phi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)\right)$$
4	2 3	ax-mp	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\chi \to \phi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)$$