Theorem re1ax2lem 32382

Description: Lemma for re1ax2 32383. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
re1ax2lem	|- ( ( ph -> ( ps -> ch ) ) -> ( ps -> ( ph -> ch ) ) )

Proof of Theorem re1ax2lem

Step	Hyp	Ref	Expression
1		tb-ax2 32379	. . . 4 |- ( ps -> ( ( ps -> ch ) -> ps ) )
2		tb-ax1 32378	. . . 4 |- ( ( ( ps -> ch ) -> ps ) -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
3	1, 2	tbsyl 32381	. . 3 |- ( ps -> ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
4		tb-ax1 32378	. . . 4 |- ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) )
5		tb-ax3 32380	. . . 4 |- ( ( ( ( ( ps -> ch ) -> ch ) -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
6	4, 5	tbsyl 32381	. . 3 |- ( ( ( ps -> ch ) -> ( ( ps -> ch ) -> ch ) ) -> ( ( ps -> ch ) -> ch ) )
7	3, 6	tbsyl 32381	. 2 |- ( ps -> ( ( ps -> ch ) -> ch ) )
8		tb-ax1 32378	. 2 |- ( ( ph -> ( ps -> ch ) ) -> ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) )
9		tb-ax1 32378	. 2 |- ( ( ps -> ( ( ps -> ch ) -> ch ) ) -> ( ( ( ( ps -> ch ) -> ch ) -> ( ph -> ch ) ) -> ( ps -> ( ph -> ch ) ) ) )
10	7, 8, 9	mpsyl 68	1 |- ( ( ph -> ( ps -> ch ) ) -> ( 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:  re1ax2  32383