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  3153  un12  3363  in12  3416  indif1  3450  difundir  3458  difindir  3460  dif32  3468  resmpt3  5060  xp0  5154  fvsnun1  5846  caov12  6206  caov13  6208  djuassen  7422  xpdjuen  7423  rec1nq  7605  halfnqq  7620  negsubdii  8454  halfpm6th  9354  decmul1  9664  i4  10894  fac4  10985  imi  11451  resqrexlemover  11561  ef01bndlem  12307  modsubi  12982  gcdmodi  12984  numexpp1  12987  karatsuba  12993  znnen  13009  sn0cld  14851  cospi  15514  sincos4thpi  15554  sincos3rdpi  15557  lgsdir2lem1  15747  lgsdir2lem5  15751  2lgsoddprmlem3d  15829  ex-bc  16261  ex-gcd  16263