Theorem minimp-pm2.43 1565

Description: Derivation of pm2.43 56 (also called "hilbert" or W) from ax-mp 5 and minimp 1560. It uses the classical derivation from ax-1 6 and ax-2 7 written DD22D21 in D-notation (see head comment for an explanation) and shortens the proof using mp2 9 (which only requires ax-mp 5). (Contributed by BJ, 31-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem minimp-pm2.43

Step	Hyp	Ref	Expression
1		minimp-ax2 1564	. 2 |- ( ( ph -> ( ph -> ps ) ) -> ( ( ph -> ph ) -> ( ph -> ps ) ) )
2		minimp-ax1 1562	. . 3 |- ( ph -> ( ( ph -> ps ) -> ph ) )
3		minimp-ax2 1564	. . 3 |- ( ( ph -> ( ( ph -> ps ) -> ph ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ph ) ) )
4	2, 3	ax-mp 5	. 2 |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ph ) )
5		minimp-ax2 1564	. 2 |- ( ( ( ph -> ( ph -> ps ) ) -> ( ( ph -> ph ) -> ( ph -> ps ) ) ) -> ( ( ( ph -> ( ph -> ps ) ) -> ( ph -> ph ) ) -> ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) ) )
6	1, 4, 5	mp2 9	1 |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by: (None)