Theorem frege60c

Description: Swap antecedents of frege58c . Proposition 60 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege59c.a	$$⊢A\in B$$
Assertion	frege60c	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	$$⊢A\in B$$
2	1	frege58c	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)$$
3		sbcim1	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\right)$$
4		sbcim1	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\psi \to \chi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)$$
5	3 4	syl6	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$
6	2 5	syl	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$
7		frege12	$$⊢\left(\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)\to \left(\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)\right)$$
8	6 7	ax-mp	$$⊢\forall x\left(\phi \to \left(\psi \to \chi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\chi \right)\right)$$