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 ) )

Syntax hints:   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  3bitr3g  222  imbibi  252  ianordc  899  19.16  1555  19.19  1666  cbvab  2301  necon1bbiidc  2408  rspc2gv  2855  elabgt  2880  sbceq1a  2974  sbcralt  3041  sbcrext  3042  sbccsbg  3088  sbccsb2g  3089  iunpw  4482  tfis  4584  xp11m  5069  ressn  5171  fnssresb  5330  fun11iun  5484  funimass4  5569  dffo4  5667  f1ompt  5670  fliftf  5803  resoprab2  5975  ralrnmpo  5992  rexrnmpo  5993  1stconst  6225  2ndconst  6226  dfsmo2  6291  smoiso  6306  brecop  6628  ixpsnf1o  6739  ac6sfi  6901  ismkvnex  7156  nninfwlporlemd  7173  prarloclemn  7501  axcaucvglemres  7901  reapti  8539  indstr  9596  iccneg  9992  sqap0  10590  sqrt00  11052  minclpr  11248  fprodseq  11594  absefib  11781  efieq1re  11782  prmind2  12123  eqgval  13088  sincosq3sgn  14389  sincosq4sgn  14390  lgsdinn0  14589  pw1nct  14893  iswomninnlem  14938