Theorem 3imtr3i 256

Description: A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)

Hypotheses

Ref	Expression
3imtr3.1	|- (ph -> ps)
3imtr3.2	|- (ph <-> ch)
3imtr3.3	|- (ps <-> th)

Assertion

Ref	Expression
3imtr3i	|- (ch -> th)

Proof of Theorem 3imtr3i

Step	Hyp	Ref	Expression
1		3imtr3.2	. . 3 |- (ph <-> ch)
2		3imtr3.1	. . 3 |- (ph -> ps)
3	1, 2	sylbir 204	. 2 |- (ch -> ps)
4		3imtr3.3	. 2 |- (ps <-> th)
5	3, 4	sylib 188	1 |- (ch -> th)

Syntax hints:   -> wi 4   <-> wb 176

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 177

This theorem is referenced by:  rb-ax1  1517  speimfw  1645  19.8wOLD  1693  ax10lem2  1937  sbal  2127  hblem  2458  fvfullfunlem2  5863  idssen  6041