Theorem 3imtr3d 202

Description: More general version of 3imtr3i 200. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)

Hypotheses

Ref	Expression
3imtr3d.1	|- ( ph -> ( ps -> ch ) )
3imtr3d.2	|- ( ph -> ( ps <-> th ) )
3imtr3d.3	|- ( ph -> ( ch <-> ta ) )

Assertion

Ref	Expression
3imtr3d	|- ( ph -> ( th -> ta ) )

Proof of Theorem 3imtr3d

Step	Hyp	Ref	Expression
1		3imtr3d.2	. 2 |- ( ph -> ( ps <-> th ) )
2		3imtr3d.1	. . 3 |- ( ph -> ( ps -> ch ) )
3		3imtr3d.3	. . 3 |- ( ph -> ( ch <-> ta ) )
4	2, 3	sylibd 149	. 2 |- ( ph -> ( ps -> ta ) )
5	1, 4	sylbird 170	1 |- ( ph -> ( th -> ta ) )

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:  f1imass  5904  focdmex  6266  tposfn2  6418  eroveu  6781  ismkvnex  7333  indpi  7540  axcaucvglemres  8097  qsqeqor  10884  caucvgrelemcau  11506  m1dvdsndvds  12786  pcpremul  12831  pcaddlem  12877  pockthlem  12894  issgrpd  13460  ghmf1  13825  islssmd  14338  znrrg  14639  limccnpcntop  15364  sincosq1sgn  15515  sincosq2sgn  15516  lgseisenlem2  15765  subctctexmid  16425  neap0mkv  16497