Theorem bitr3id 194

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

Hypotheses

Ref	Expression
bitr3id.1	⊢ (𝜓 ↔ 𝜑)
bitr3id.2	⊢ (𝜒 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
bitr3id	⊢ (𝜒 → (𝜑 ↔ 𝜃))

Proof of Theorem bitr3id

Step	Hyp	Ref	Expression
1		bitr3id.1	. . 3 ⊢ (𝜓 ↔ 𝜑)
2	1	bicomi 132	. 2 ⊢ (𝜑 ↔ 𝜓)
3		bitr3id.2	. 2 ⊢ (𝜒 → (𝜓 ↔ 𝜃))
4	2, 3	bitrid 192	1 ⊢ (𝜒 → (𝜑 ↔ 𝜃))

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  7501  nninfwlporlemd  7518  prarloclemn  7872  axcaucvglemres  8272  reapti  8915  indstr  10008  iccneg  10407  sqap0  11063  wrdmap  11357  wrdind  11515  sqrt00  11827  minclpr  12026  fprodseq  12374  absefib  12562  efieq1re  12563  prmind2  12922  ballotfilemsima  13316  gzsumval2  13772  eqgval  14084  resscntz  14165  isnzr2  14599  sincosq3sgn  16072  sincosq4sgn  16073  fsumdvdsmul  16302  ppiqub  16310  lgsdinn0  16389  pw1nct  17255  iswomninnlem  17323