Metamath Proof Explorer

Theorem frege56a

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

		Ref	Expression
	Assertion	frege56a	\|- ( ( ( ph <-> ps ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) -> ( ( ps <-> ph ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege55cor1a	\|- ( ( ps <-> ph ) -> ( ph <-> ps ) )
2		frege9	\|- ( ( ( ps <-> ph ) -> ( ph <-> ps ) ) -> ( ( ( ph <-> ps ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) -> ( ( ps <-> ph ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) ) )
3	1 2	ax-mp	\|- ( ( ( ph <-> ps ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) -> ( ( ps <-> ph ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) )

Step

Hyp

Ref

Expression

frege55cor1a

 |-  ( ( ps <-> ph ) -> ( ph <-> ps ) )

frege9

 |-  ( ( ( ps <-> ph ) -> ( ph <-> ps ) ) -> ( ( ( ph <-> ps ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) -> ( ( ps <-> ph ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) ) )

1 2

ax-mp

 |-  ( ( ( ph <-> ps ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) -> ( ( ps <-> ph ) -> ( if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) )