Theorem 3eqtr3i 2260

Description: An inference from three chained equalities. (Contributed by NM, 6-May-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
3eqtr3i.1	|- A = B
3eqtr3i.2	|- A = C
3eqtr3i.3	|- B = D

Assertion

Ref	Expression
3eqtr3i	|- C = D

Proof of Theorem 3eqtr3i

Step	Hyp	Ref	Expression
1		3eqtr3i.1	. . 3 |- A = B
2		3eqtr3i.2	. . 3 |- A = C
3	1, 2	eqtr3i 2254	. 2 |- B = C
4		3eqtr3i.3	. 2 |- B = D
5	3, 4	eqtr3i 2254	1 |- C = D

Syntax hints:   = wceq 1398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  csbvarg  3156  un12  3367  in12  3420  indif1  3454  difundir  3462  difindir  3464  dif32  3472  resmpt3  5068  xp0  5163  fvsnun1  5859  caov12  6221  caov13  6223  djuassen  7492  xpdjuen  7493  rec1nq  7675  halfnqq  7690  negsubdii  8523  halfpm6th  9423  decmul1  9735  i4  10967  fac4  11058  imi  11540  resqrexlemover  11650  ef01bndlem  12397  modsubi  13072  gcdmodi  13074  numexpp1  13077  karatsuba  13083  znnen  13099  sn0cld  14948  cospi  15611  sincos4thpi  15651  sincos3rdpi  15654  lgsdir2lem1  15847  lgsdir2lem5  15851  2lgsoddprmlem3d  15929  ex-bc  16443  ex-gcd  16445