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  5053  xp0  5147  fvsnun1  5835  caov12  6193  caov13  6195  djuassen  7395  xpdjuen  7396  rec1nq  7578  halfnqq  7593  negsubdii  8427  halfpm6th  9327  decmul1  9637  i4  10859  fac4  10950  imi  11406  resqrexlemover  11516  ef01bndlem  12262  modsubi  12937  gcdmodi  12939  numexpp1  12942  karatsuba  12948  znnen  12964  sn0cld  14805  cospi  15468  sincos4thpi  15508  sincos3rdpi  15511  lgsdir2lem1  15701  lgsdir2lem5  15705  2lgsoddprmlem3d  15783  ex-bc  16051  ex-gcd  16053