Theorem 3imtr3d 201

Description: More general version of 3imtr3i 199. 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 148	. 2 |- ( ph -> ( ps -> ta ) )
5	1, 4	sylbird 169	1 |- ( ph -> ( th -> ta ) )

Syntax hints:   -> wi 4   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  f1imass  5668  fornex  6006  tposfn2  6156  eroveu  6513  ismkvnex  7022  indpi  7143  axcaucvglemres  7700  caucvgrelemcau  10745  limccnpcntop  12802  sincosq1sgn  12896  sincosq2sgn  12897  subctctexmid  13185