Theorem 3eqtr3i 2258

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 2252	. 2 |- B = C
4		3eqtr3i.3	. 2 |- B = D
5	3, 4	eqtr3i 2252	1 |- C = D

Syntax hints:   = wceq 1395

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  csbvarg  3152  un12  3362  in12  3415  indif1  3449  difundir  3457  difindir  3459  dif32  3467  resmpt3  5054  xp0  5148  fvsnun1  5836  caov12  6194  caov13  6196  djuassen  7399  xpdjuen  7400  rec1nq  7582  halfnqq  7597  negsubdii  8431  halfpm6th  9331  decmul1  9641  i4  10864  fac4  10955  imi  11411  resqrexlemover  11521  ef01bndlem  12267  modsubi  12942  gcdmodi  12944  numexpp1  12947  karatsuba  12953  znnen  12969  sn0cld  14811  cospi  15474  sincos4thpi  15514  sincos3rdpi  15517  lgsdir2lem1  15707  lgsdir2lem5  15711  2lgsoddprmlem3d  15789  ex-bc  16093  ex-gcd  16095