Metamath Proof Explorer

Theorem frege62b

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
	Assertion	frege62b	\|- ( [ y / x ] ph -> ( A. x ( ph -> ps ) -> [ y / x ] ps ) )

Proof

Step	Hyp	Ref	Expression
1		frege58bcor	\|- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
2		ax-frege8	\|- ( ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) ) -> ( [ y / x ] ph -> ( A. x ( ph -> ps ) -> [ y / x ] ps ) ) )
3	1 2	ax-mp	\|- ( [ y / x ] ph -> ( A. x ( ph -> ps ) -> [ y / x ] ps ) )

Step

Hyp

Ref

Expression

frege58bcor

 |-  ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )

ax-frege8

 |-  ( ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) ) -> ( [ y / x ] ph -> ( A. x ( ph -> ps ) -> [ y / x ] ps ) ) )

1 2

ax-mp

 |-  ( [ y / x ] ph -> ( A. x ( ph -> ps ) -> [ y / x ] ps ) )