Metamath Proof Explorer

Theorem rp-frege4g

Description: Deduction related to distribution. (Contributed by RP, 24-Dec-2019)

		Ref	Expression
	Assertion	rp-frege4g	\|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		rp-frege3g	\|- ( ph -> ( ( ps -> ( ch -> th ) ) -> ( ( ps -> ch ) -> ( ps -> th ) ) ) )
2		ax-frege2	\|- ( ( ph -> ( ( ps -> ( ch -> th ) ) -> ( ( ps -> ch ) -> ( ps -> th ) ) ) ) -> ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) ) )
3	1 2	ax-mp	\|- ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) )

Step

Hyp

Ref

Expression

rp-frege3g

 |-  ( ph -> ( ( ps -> ( ch -> th ) ) -> ( ( ps -> ch ) -> ( ps -> th ) ) ) )

ax-frege2

 |-  ( ( ph -> ( ( ps -> ( ch -> th ) ) -> ( ( ps -> ch ) -> ( ps -> th ) ) ) ) -> ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) ) )

1 2

ax-mp

 |-  ( ( ph -> ( ps -> ( ch -> th ) ) ) -> ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) ) )