Metamath Proof Explorer


Theorem frege62c

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara when the minor premise has a particular context. Proposition 62 of Frege1879 p. 52. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

Ref Expression
Hypothesis frege59c.a A B
Assertion frege62c [˙A / x]˙ φ x φ ψ [˙A / x]˙ 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 ax-frege8 x φ ψ [˙A / x]˙ φ [˙A / x]˙ ψ [˙A / x]˙ φ x φ ψ [˙A / x]˙ ψ
6 4 5 ax-mp [˙A / x]˙ φ x φ ψ [˙A / x]˙ ψ