Theorem re1ax2 32383

Description: ax-2 7 rederived from the Tarski-Bernays axiom system. Often tb-ax1 32378 is replaced with this theorem to make a "standard" system. This is because this theorem is easier to work with, despite it being longer. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1ax2	|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )

Proof of Theorem re1ax2

Step	Hyp	Ref	Expression
1		re1ax2lem 32382	. 2 |- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )
2		tb-ax1 32378	. . . 4 |- ( ( ph -> ( ph -> ch ) ) -> ( ( ( ph -> ch ) -> ch ) -> ( ph -> ch ) ) )
3		tb-ax3 32380	. . . 4 |- ( ( ( ( ph -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ph -> ch ) )
4	2, 3	tbsyl 32381	. . 3 |- ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) )
5		tb-ax1 32378	. . . 4 |- ( ( ph -> ps ) -> ( ( ps -> ( ph -> ch ) ) -> ( ph -> ( ph -> ch ) ) ) )
6		re1ax2lem 32382	. . . 4 |- ( ( ( ph -> ps ) -> ( ( ps -> ( ph -> ch ) ) -> ( ph -> ( ph -> ch ) ) ) ) -> ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) ) )
7	5, 6	ax-mp 5	. . 3 |- ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) )
8		tb-ax1 32378	. . . 4 |- ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
9		re1ax2lem 32382	. . . 4 |- ( ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) -> ( ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) ) -> ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) ) )
10	8, 9	ax-mp 5	. . 3 |- ( ( ( ph -> ( ph -> ch ) ) -> ( ph -> ch ) ) -> ( ( ( ph -> ps ) -> ( ph -> ( ph -> ch ) ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) ) )
11	4, 7, 10	mpsyl 68	. 2 |- ( ( ps -> ( ph -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )
12	1, 11	tbsyl 32381	1 |- ( ( ph -> ( ps -> ch ) ) -> ( ( ph -> ps ) -> ( ph -> ch ) ) )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by: (None)