Metamath Proof Explorer

Theorem frege68a

Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of Frege1879 p. 54. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	frege68a	\|- ( ( ( ps /\ ch ) <-> th ) -> ( th -> if- ( ph , ps , ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege57aid	\|- ( ( ( ps /\ ch ) <-> th ) -> ( th -> ( ps /\ ch ) ) )
2		frege67a	\|- ( ( ( ( ps /\ ch ) <-> th ) -> ( th -> ( ps /\ ch ) ) ) -> ( ( ( ps /\ ch ) <-> th ) -> ( th -> if- ( ph , ps , ch ) ) ) )
3	1 2	ax-mp	\|- ( ( ( ps /\ ch ) <-> th ) -> ( th -> if- ( ph , ps , ch ) ) )

Step

Hyp

Ref

Expression

frege57aid

 |-  ( ( ( ps /\ ch ) <-> th ) -> ( th -> ( ps /\ ch ) ) )

frege67a

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

1 2

ax-mp

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