Theorem bitr3VD 39084

Description: Virtual deduction proof of bitr3 342. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	|- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2:1,?: e1a 38852	|- (. ( ph <-> ps ) ->. ( ps <-> ph ) ).
3::	|- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ph <-> ch ) ).
4:3,?: e2 38856	|- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ch <-> ph ) ).
5:2,4,?: e12 38951	|- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ps <-> ch ) ).
6:5:	|- (. ( ph <-> ps ) ->. ( ( ph <-> ch ) -> ( ps <-> ch ) ) ).
qed:6:	|- ( ( ph <-> ps ) -> ( ( ph <-> ch ) -> ( ps <-> ch ) ) )

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bitr3VD	|- ( ( ph <-> ps ) -> ( ( ph <-> ch ) -> ( ps <-> ch ) ) )

Proof of Theorem bitr3VD

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( ( ph <-> ps ) -> ( ph <-> ps ) )
2	1	bicomd 213	. 2 |- ( ( ph <-> ps ) -> ( ps <-> ph ) )
3		id 22	. . 3 |- ( ( ph <-> ch ) -> ( ph <-> ch ) )
4	3	bicomd 213	. 2 |- ( ( ph <-> ch ) -> ( ch <-> ph ) )
5		biantr 972	. . 3 |- ( ( ( ps <-> ph ) /\ ( ch <-> ph ) ) -> ( ps <-> ch ) )
6	5	ex 450	. 2 |- ( ( ps <-> ph ) -> ( ( ch <-> ph ) -> ( ps <-> ch ) ) )
7	2, 4, 6	syl2im 40	1 |- ( ( ph <-> ps ) -> ( ( ph <-> ch ) -> ( ps <-> ch ) ) )

Syntax hints:   -> 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  df-an 386

This theorem is referenced by: (None)