Metamath Proof Explorer

Theorem frege64b

Description: Lemma for frege65b . Proposition 64 of Frege1879 p. 53. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege64b	\|- ( ( [ x / y ] ph -> [ z / y ] ps ) -> ( A. y ( ps -> ch ) -> ( [ x / y ] ph -> [ z / y ] ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege62b	\|- ( [ z / y ] ps -> ( A. y ( ps -> ch ) -> [ z / y ] ch ) )
2		frege18	\|- ( ( [ z / y ] ps -> ( A. y ( ps -> ch ) -> [ z / y ] ch ) ) -> ( ( [ x / y ] ph -> [ z / y ] ps ) -> ( A. y ( ps -> ch ) -> ( [ x / y ] ph -> [ z / y ] ch ) ) ) )
3	1 2	ax-mp	\|- ( ( [ x / y ] ph -> [ z / y ] ps ) -> ( A. y ( ps -> ch ) -> ( [ x / y ] ph -> [ z / y ] ch ) ) )

Step

Hyp

Ref

Expression

frege62b

 |-  ( [ z / y ] ps -> ( A. y ( ps -> ch ) -> [ z / y ] ch ) )

frege18

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

1 2

ax-mp

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