Theorem 3eqtr3i 2267

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 2261	. 2 |- B = C
4		3eqtr3i.3	. 2 |- B = D
5	3, 4	eqtr3i 2261	1 |- C = D

This proof depends on syntax axioms:   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  csbvarg  3175  un12  3387  in12  3442  indif1  3476  difundir  3484  difindir  3486  dif32  3494  resmpt3  5115  xp0  5210  fvsnun1  5915  caov12  6281  caov13  6283  djuassen  7579  xpdjuen  7580  rec1nq  7768  halfnqq  7783  negsubdii  8618  halfpm6th  9536  decmul1  9856  i4  11100  fac4  11193  imi  11688  resqrexlemover  11798  ef01bndlem  12548  modsubi  13228  gcdmodi  13230  numexpp1  13233  karatsuba  13239  ballotfilemth  13339  znnen  13347  sn0cld  15363  cospi  16027  sincos4thpi  16067  sincos3rdpi  16070  log2ublem2  16222  log2ublog2  16224  chtqub  16296  bclbnd  16307  bposlem8  16318  lgsdir2lem1  16352  lgsdir2lem5  16356  2lgsoddprmlem3d  16434  ex-bc  16948  ex-gcd  16950