Theorem 3eqtr3i 2263

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 2257	. 2 ⊢ 𝐵 = 𝐶
4		3eqtr3i.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	eqtr3i 2257	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 2216

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  csbvarg  3169  un12  3381  in12  3436  indif1  3470  difundir  3478  difindir  3480  dif32  3488  resmpt3  5092  xp0  5187  fvsnun1  5886  caov12  6251  caov13  6253  djuassen  7537  xpdjuen  7538  rec1nq  7726  halfnqq  7741  negsubdii  8575  halfpm6th  9478  decmul1  9793  i4  11031  fac4  11123  imi  11613  resqrexlemover  11723  ef01bndlem  12470  modsubi  13145  gcdmodi  13147  numexpp1  13150  karatsuba  13156  ballotfilemth  13228  znnen  13236  sn0cld  15131  cospi  15794  sincos4thpi  15834  sincos3rdpi  15837  lgsdir2lem1  16030  lgsdir2lem5  16034  2lgsoddprmlem3d  16112  ex-bc  16626  ex-gcd  16628