Metamath Proof Explorer

Theorem merco1lem15

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 18-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	merco1lem15	\|- ( ( ph -> ps ) -> ( ph -> ( ch -> ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		merco1lem14	\|- ( ( ( ( ph -> ps ) -> ps ) -> ( ch -> ps ) ) -> ( ph -> ( ch -> ps ) ) )
2		merco1lem13	\|- ( ( ( ( ( ph -> ps ) -> ps ) -> ( ch -> ps ) ) -> ( ph -> ( ch -> ps ) ) ) -> ( ( ph -> ps ) -> ( ph -> ( ch -> ps ) ) ) )
3	1 2	ax-mp	\|- ( ( ph -> ps ) -> ( ph -> ( ch -> ps ) ) )

Step

Hyp

Ref

Expression

merco1lem14

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

merco1lem13

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

1 2

ax-mp

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