Theorem 3imtr3g 260

Description: More general version of 3imtr3i 256. 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 209	. 2 |- (ph -> (th -> ch))
4		3imtr3g.3	. 2 |- (ch <-> ta)
5	3, 4	syl6ib 217	1 |- (ph -> (th -> ta))

Syntax hints:   -> wi 4   <-> wb 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  exim  1575  dvelimhw  1849  ax12olem2  1928  dvelimh  1964  dvelimALT  2133  dvelimf-o  2180  axi11e  2332  sspwb  4118  pw1disj  4167  ssopab2b  4713  imadif  5171  enpw1  6062