Theorem frege59c

Description: A kind of Aristotelian inference. Proposition 59 of Frege1879 p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collectionFrom Frege to Goedel, this proof has the frege12 incorrectly referenced where frege30 is in the original. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		frege59c.a	$$⊢A\in B$$
2	1	frege58c	$$⊢\forall x\left(\phi \to \psi \right)\to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)$$
3		sbcim1	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$
4	2 3	syl	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)$$
5		frege30	$$⊢\left(\forall x\left(\phi \to \psi \right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \right)\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\neg \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \neg \forall x\left(\phi \to \psi \right)\right)\right)$$
6	4 5	ax-mp	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \to \left(\neg \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\psi \to \neg \forall x\left(\phi \to \psi \right)\right)$$