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  5112  xp0  5207  fvsnun1  5912  caov12  6278  caov13  6280  djuassen  7574  xpdjuen  7575  rec1nq  7763  halfnqq  7778  negsubdii  8613  halfpm6th  9530  decmul1  9850  i4  11094  fac4  11187  imi  11682  resqrexlemover  11792  ef01bndlem  12542  modsubi  13222  gcdmodi  13224  numexpp1  13227  karatsuba  13233  ballotfilemth  13333  znnen  13341  sn0cld  15329  cospi  15993  sincos4thpi  16033  sincos3rdpi  16036  log2ublem2  16183  log2ublog2  16185  chtqub  16257  bclbnd  16268  bposlem8  16279  lgsdir2lem1  16313  lgsdir2lem5  16317  2lgsoddprmlem3d  16395  ex-bc  16909  ex-gcd  16911