Theorem 3eqtr3i 2261

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 2255	. 2 ⊢ 𝐵 = 𝐶
4		3eqtr3i.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	eqtr3i 2255	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 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225

This theorem is referenced by:  csbvarg  3165  un12  3376  in12  3429  indif1  3463  difundir  3471  difindir  3473  dif32  3481  resmpt3  5078  xp0  5173  fvsnun1  5872  caov12  6234  caov13  6236  djuassen  7515  xpdjuen  7516  rec1nq  7698  halfnqq  7713  negsubdii  8546  halfpm6th  9446  decmul1  9758  i4  10990  fac4  11081  imi  11563  resqrexlemover  11673  ef01bndlem  12420  modsubi  13095  gcdmodi  13097  numexpp1  13100  karatsuba  13106  znnen  13123  sn0cld  14972  cospi  15635  sincos4thpi  15675  sincos3rdpi  15678  lgsdir2lem1  15871  lgsdir2lem5  15875  2lgsoddprmlem3d  15953  ex-bc  16467  ex-gcd  16469