Theorem bitr3 1283

Description: Closed nested implication form of bitr3i 192. Derived automatically from bitr3VD 16673. (Contributed by Alan Sare, 31-Dec-2011.)

Assertion

Ref	Expression
bitr3	|- ((ph <-> ps) -> ((ph <-> ch) -> (ps <-> ch)))

Proof of Theorem bitr3

Step	Hyp	Ref	Expression
1		id 73	. . 3 |- ((ph <-> ps) -> (ph <-> ps))
2	1	bicomd 580	. 2 |- ((ph <-> ps) -> (ps <-> ph))
3		id 73	. . . 4 |- ((ph <-> ch) -> (ph <-> ch))
4	3	bicomd 580	. . 3 |- ((ph <-> ch) -> (ch <-> ph))
5	4	a1i 8	. 2 |- ((ph <-> ps) -> ((ph <-> ch) -> (ch <-> ph)))
6		biantr 814	. 2 |- (((ps <-> ph) /\ (ch <-> ph)) -> (ps <-> ch))
7	2, 5, 6	ee12an 1273	1 |- ((ph <-> ps) -> ((ph <-> ch) -> (ps <-> ch)))

Syntax hints:   -> wi 3   <-> wb 163

This theorem is referenced by:  3orbi123VD 16674  sbc3orgVD 16675  trsbcVD 16701

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-an 242

Copyright terms: Public domain