Metamath Proof Explorer

Theorem frege56b

Description: Lemma for frege57b . Proposition 56 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege56b	\|- ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) -> ( y = x -> ( [ x / z ] ph -> [ y / z ] ph ) ) )

Step	Hyp	Ref	Expression
1		frege55b	\|- ( y = x -> x = y )
2		frege9	\|- ( ( y = x -> x = y ) -> ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) -> ( y = x -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) )
3	1 2	ax-mp	\|- ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) -> ( y = x -> ( [ x / z ] ph -> [ y / z ] ph ) ) )

Step

Hyp

Ref

Expression

 |-  ( y = x -> x = y )

 |-  ( ( y = x -> x = y ) -> ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) -> ( y = x -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) )

1 2

 |-  ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) -> ( y = x -> ( [ x / z ] ph -> [ y / z ] ph ) ) )