Theorem syl7bi 245

Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.)

Hypotheses

Ref	Expression
syl7bi.1	|- ( ph <-> ps )
syl7bi.2	|- ( ch -> ( th -> ( ps -> ta ) ) )

Assertion

Ref	Expression
syl7bi	|- ( ch -> ( th -> ( ph -> ta ) ) )

Proof of Theorem syl7bi

Step	Hyp	Ref	Expression
1		syl7bi.1	. . 3 |- ( ph <-> ps )
2	1	biimpi 206	. 2 |- ( ph -> ps )
3		syl7bi.2	. 2 |- ( ch -> ( th -> ( ps -> ta ) ) )
4	2, 3	syl7 74	1 |- ( ch -> ( th -> ( ph -> ta ) ) )

Syntax hints:   -> wi 4   <-> wb 196

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 197

This theorem is referenced by:  nfimt  1821  rspct  3302  zfpair  4904  gruen  9634  axpre-sup  9990  nn0lt2  11440  fzofzim  12514  ndvdssub  15133  alexsubALT  21855  clwlkclwwlklem2a  26899  erclwwlkstr  26936  erclwwlksntr  26948  dfon2lem8  31695  prtlem15  34160  prtlem18  34162