Theorem 3eqtr3i 2222

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 2216	. 2 |- B = C
4		3eqtr3i.3	. 2 |- B = D
5	3, 4	eqtr3i 2216	1 |- C = D

Syntax hints:   = wceq 1364

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186

This theorem is referenced by:  csbvarg  3108  un12  3317  in12  3370  indif1  3404  difundir  3412  difindir  3414  dif32  3422  resmpt3  4991  xp0  5085  fvsnun1  5755  caov12  6107  caov13  6109  djuassen  7277  xpdjuen  7278  rec1nq  7455  halfnqq  7470  negsubdii  8304  halfpm6th  9202  decmul1  9511  i4  10713  fac4  10804  imi  11044  resqrexlemover  11154  ef01bndlem  11899  znnen  12555  sn0cld  14305  cospi  14935  sincos4thpi  14975  sincos3rdpi  14978  lgsdir2lem1  15144  lgsdir2lem5  15148  2lgsoddprmlem3d  15198  ex-bc  15221  ex-gcd  15223