Theorem 3imtr3g 193

Description: More general version of 3imtr3i 189. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)

Hypotheses

Ref	Expression
3imtr3g.1	|- ( ph -> ( ps -> ch ) )
3imtr3g.2	|- ( ps <-> th )
3imtr3g.3	|- ( ch <-> ta )

Assertion

Ref	Expression
3imtr3g	|- ( ph -> ( th -> ta ) )

Proof of Theorem 3imtr3g

Step	Hyp	Ref	Expression
1		3imtr3g.2	. . 3 |- ( ps <-> th )
2		3imtr3g.1	. . 3 |- ( ph -> ( ps -> ch ) )
3	1, 2	syl5bir 142	. 2 |- ( ph -> ( th -> ch ) )
4		3imtr3g.3	. 2 |- ( ch <-> ta )
5	3, 4	syl6ib 150	1 |- ( ph -> ( th -> ta ) )

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:  dvelimfALT2  1698  dvelimf  1891  dveeq1  1895  sspwb  3952  ssopab2b  4013  wetrep  4097  imadif  4979  ssoprab2b  5562  iinerm  6178  uzind  8349