Theorem pm2.43cbi 38724

Description: Logical equivalence of a 3-left-nested implication and a 2-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.

1::	|- ( ( ph -> ( ps -> ( ch -> ( ph -> th ) ) ) ) -> ( ph -> ( ps -> ( ph -> ( ch -> th ) ) ) ) )
2::	|- ( ( ph -> ( ps -> ( ph -> ( ch -> th ) ) ) ) -> ( ps -> ( ph -> ( ch -> th ) ) ) )
3:1,2:	|- ( ( ph -> ( ps -> ( ch -> ( ph -> th ) ) ) ) -> ( ps -> ( ph -> ( ch -> th ) ) ) )
4::	|- ( ( ps -> ( ph -> ( ch -> th ) ) ) -> ( ps -> ( ch -> ( ph -> th ) ) ) )
5:3,4:	|- ( ( ph -> ( ps -> ( ch -> ( ph -> th ) ) ) ) -> ( ps -> ( ch -> ( ph -> th ) ) ) )
6::	|- ( ( ps -> ( ch -> ( ph -> th ) ) ) -> ( ph -> ( ps -> ( ch -> ( ph -> th ) ) ) ) )
qed:5,6:	|- ( ( ph -> ( ps -> ( ch -> ( ph -> th ) ) ) ) <-> ( ps -> ( ch -> ( ph -> th ) ) ) )

Assertion

Ref	Expression
pm2.43cbi	|- ( ( ph -> ( ps -> ( ch -> ( ph -> th ) ) ) ) <-> ( ps -> ( ch -> ( ph -> th ) ) ) )

Proof of Theorem pm2.43cbi

Step	Hyp	Ref	Expression
1		pm2.24 121	. . . 4 |- ( ph -> ( -. ph -> ( ps -> ( ch -> th ) ) ) )
2	1	com4l 92	. . 3 |- ( -. ph -> ( ps -> ( ch -> ( ph -> th ) ) ) )
3		id 22	. . 3 |- ( ( ps -> ( ch -> ( ph -> th ) ) ) -> ( ps -> ( ch -> ( ph -> th ) ) ) )
4	2, 3	ja 173	. 2 |- ( ( ph -> ( ps -> ( ch -> ( ph -> th ) ) ) ) -> ( ps -> ( ch -> ( ph -> th ) ) ) )
5		ax-1 6	. 2 |- ( ( ps -> ( ch -> ( ph -> th ) ) ) -> ( ph -> ( ps -> ( ch -> ( ph -> th ) ) ) ) )
6	4, 5	impbii 199	1 |- ( ( ph -> ( ps -> ( ch -> ( ph -> th ) ) ) ) <-> ( ps -> ( ch -> ( ph -> th ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196

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

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  ee233  38725  ee33VD  39115