Theorem 3imtr3i 189

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 125	. 2 |- ( ch -> ps )
4		3imtr3.3	. 2 |- ( ps <-> th )
5	3, 4	sylib 127	1 |- ( ch -> th )

Syntax hints:   -> wi 4   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  cbv1  1632  moimv  1966  hblem  2145  tfi  4305  smores  5907  idssen  6257