Metamath Proof Explorer

Theorem frege58acor

Description: Lemma for frege59a . (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege58acor	\|- ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> ( if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) )

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

Step

Hyp

Ref

Expression

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

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

1 2

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