Theorem 3imtr4d 421

Description: More general version of 3imtr4 192. Useful for converting conditional definitions in a formula.

Hypotheses

Ref	Expression
3imtr4d.1	|- (ph -> (ps -> ch))
3imtr4d.2	|- (ph -> (th <-> ps))
3imtr4d.3	|- (ph -> (ta <-> ch))

Assertion

Ref	Expression
3imtr4d	|- (ph -> (th -> ta ))

Proof of Theorem 3imtr4d

Step	Hyp	Ref	Expression
1		3imtr4d.2	. 2 |- (ph -> (th <-> ps))
2		3imtr4d.1	. . 3 |- (ph -> (ps -> ch))
3		3imtr4d.3	. . 3 |- (ph -> (ta <-> ch))
4	2, 3	sylibrd 179	. 2 |- (ph -> (ps -> ta ))
5	1, 4	sylbid 178	1 |- (ph -> (th -> ta ))

Syntax hints:   -> wi 2   <-> wb 127

This theorem is referenced by:  oaord 3149  nnmord 3189  nnmcan 3190  omsmo 3196  oprec 3254  ltsopi 3810  nnge1t 4439  znnen 4930  ocsh 5164  spansncv2t 5725

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

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