Metamath Proof Explorer


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 B
Assertion frege59c [˙A / x]˙ φ ¬ [˙A / x]˙ ψ ¬ x φ ψ

Proof

Step Hyp Ref Expression
1 frege59c.a A B
2 1 frege58c x φ ψ [˙A / x]˙ φ ψ
3 sbcim1 [˙A / x]˙ φ ψ [˙A / x]˙ φ [˙A / x]˙ ψ
4 2 3 syl x φ ψ [˙A / x]˙ φ [˙A / x]˙ ψ
5 frege30 x φ ψ [˙A / x]˙ φ [˙A / x]˙ ψ [˙A / x]˙ φ ¬ [˙A / x]˙ ψ ¬ x φ ψ
6 4 5 ax-mp [˙A / x]˙ φ ¬ [˙A / x]˙ ψ ¬ x φ ψ