Theorem bitr3id 194

Description: A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
bitr3id.1	|- ( ps <-> ph )
bitr3id.2	|- ( ch -> ( ps <-> th ) )

Assertion

Ref	Expression
bitr3id	|- ( ch -> ( ph <-> th ) )

Proof of Theorem bitr3id

Step	Hyp	Ref	Expression
1		bitr3id.1	. . 3 |- ( ps <-> ph )
2	1	bicomi 132	. 2 |- ( ph <-> ps )
3		bitr3id.2	. 2 |- ( ch -> ( ps <-> th ) )
4	2, 3	bitrid 192	1 |- ( ch -> ( ph <-> th ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  3bitr3g  222  imbibi  252  ianordc  911  19.16  1608  19.19  1718  cbvab  2364  necon1bbiidc  2481  rspc2gv  2942  elabgt  2967  sbceq1a  3061  sbcralt  3128  sbcrext  3129  sbccsbg  3176  sbccsb2g  3177  iunpw  4628  tfis  4732  reldmm  5002  xp11m  5229  ressn  5331  fnssresb  5498  fun11iun  5663  funimass4  5756  dffo4  5859  f1ompt  5862  dfimafnf  5958  fliftf  6008  resoprab2  6188  ralrnmpo  6206  rexrnmpo  6207  1stconst  6461  2ndconst  6462  dfsmo2  6563  smoiso  6578  brecop  6904  ixpsnf1o  7024  ac6sfi  7208  ismkvnex  7502  nninfwlporlemd  7519  prarloclemn  7873  axcaucvglemres  8273  reapti  8916  indstr  10009  iccneg  10408  sqap0  11064  wrdmap  11358  wrdind  11516  sqrt00  11828  minclpr  12027  fprodseq  12375  absefib  12563  efieq1re  12564  prmind2  12923  ballotfilemsima  13317  gzsumval2  13773  eqgval  14085  resscntz  14166  isnzr2  14600  sincosq3sgn  16088  sincosq4sgn  16089  fsumdvdsmul  16318  ppiqub  16326  lgsdinn0  16405  pw1nct  17271  iswomninnlem  17339