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:  dcfromnotnotr  1493  dcfromcon  1494  dcfrompeirce  1495  cbv1  1794  cbv1v  1796  moimv  2147  hblem  2340  tfi  4703  fresaunres1disj  5545  smores  6522  idssen  7015  suplocsrlem  8122  bezoutlemle  12700  limcmpted  15520  sincosq3sgn  15685  fsumdvdsmul  15851  subctctexmid  16766  dcapnconstALT  16839