Theorem 3imtr3i 200

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 135	. 2 |- ( ch -> ps )
4		3imtr3.3	. 2 |- ( ps <-> th )
5	3, 4	sylib 122	1 |- ( ch -> th )

Syntax hints:   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  cbv1  1755  cbv1v  1757  moimv  2102  hblem  2295  tfi  4593  smores  6307  idssen  6791  suplocsrlem  7821  bezoutlemle  12023  limcmpted  14485  sincosq3sgn  14602  subctctexmid  15104  dcapnconstALT  15164