Metamath Proof Explorer

Theorem frege61a

Description: Lemma for frege65a . Proposition 61 of Frege1879 p. 52. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege61a	\|- ( ( if- ( ph , ps , ch ) -> th ) -> ( ( ps /\ ch ) -> th ) )

Step	Hyp	Ref	Expression
1		ax-frege58a	\|- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )
2		frege9	\|- ( ( ( ps /\ ch ) -> if- ( ph , ps , ch ) ) -> ( ( if- ( ph , ps , ch ) -> th ) -> ( ( ps /\ ch ) -> th ) ) )
3	1 2	ax-mp	\|- ( ( if- ( ph , ps , ch ) -> th ) -> ( ( ps /\ ch ) -> th ) )

Step

Hyp

Ref

Expression

 |-  ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )

 |-  ( ( ( ps /\ ch ) -> if- ( ph , ps , ch ) ) -> ( ( if- ( ph , ps , ch ) -> th ) -> ( ( ps /\ ch ) -> th ) ) )

1 2

 |-  ( ( if- ( ph , ps , ch ) -> th ) -> ( ( ps /\ ch ) -> th ) )