Theorem minimp-pm2.43

Description: Derivation of pm2.43 (also called "hilbert" or W) from ax-mp and minimp . It uses the classical derivation from ax-1 and ax-2 written DD22D21 in D-notation (see head comment for an explanation) and shortens the proof using mp2 (which only requires ax-mp ). (Contributed by BJ, 31-May-2021) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	minimp-pm2.43	|- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )

Proof

Step	Hyp	Ref	Expression
1		minimp-ax2	|- ( ( ph -> ( ph -> ps ) ) -> ( ( ph -> ph ) -> ( ph -> ps ) ) )
2		minimp-ax1	|- ( ph -> ( ( ph -> ps ) -> ph ) )
3		minimp-ax2	|- ( ( ph -> ( ( ph -> ps ) -> ph ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ph ) ) )
4	2 3	ax-mp	|- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ph ) )
5		minimp-ax2	|- ( ( ( ph -> ( ph -> ps ) ) -> ( ( ph -> ph ) -> ( ph -> ps ) ) ) -> ( ( ( ph -> ( ph -> ps ) ) -> ( ph -> ph ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) ) )
6	1 4 5	mp2	|- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )