Theorem 3imtr3d 420

Description: More general version of 3imtr3 191. Useful for converting conditional definitions in a formula.

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 177	. 2 |- (ph -> (ps -> ta ))
5	1, 4	sylbird 180	1 |- (ph -> (th -> ta ))

Syntax hints:   -> wi 2   <-> wb 127

This theorem is referenced by:  ddelimdf 909  fornex 2793  sdomel 3653  cardsdomel 3658  ltapr 3945  nnleltp1t 4448  pjnormss 5638

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128

metamath.org