Theorem frege65c

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara when the minor premise has a general context. Proposition 65 of Frege1879 p. 53. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	$$⊢A\in B$$
Assertion	frege65c	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	$$⊢A\in B$$
2		sbcim1	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$
3	1	frege64c	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$
4	2 3	syl	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$
5	1	frege61c	$$⊢\left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)\to \left(\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
6	4 5	ax-mp	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$