Metamath Proof Explorer

Theorem frege67a

Description: Lemma for frege68a . Proposition 67 of Frege1879 p. 54. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege67a	\|- ( ( ( ( ps /\ ch ) <-> th ) -> ( th -> ( ps /\ ch ) ) ) -> ( ( ( ps /\ ch ) <-> th ) -> ( th -> if- ( ph , ps , ch ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-frege58a	\|- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )
2		frege7	\|- ( ( ( ps /\ ch ) -> if- ( ph , ps , ch ) ) -> ( ( ( ( ps /\ ch ) <-> th ) -> ( th -> ( ps /\ ch ) ) ) -> ( ( ( ps /\ ch ) <-> th ) -> ( th -> if- ( ph , ps , ch ) ) ) ) )
3	1 2	ax-mp	\|- ( ( ( ( ps /\ ch ) <-> th ) -> ( th -> ( ps /\ ch ) ) ) -> ( ( ( ps /\ ch ) <-> th ) -> ( th -> if- ( ph , ps , ch ) ) ) )

Step

Hyp

Ref

Expression

ax-frege58a

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

frege7

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

1 2

ax-mp

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