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  5054  xp0  5148  fvsnun1  5840  caov12  6200  caov13  6202  djuassen  7410  xpdjuen  7411  rec1nq  7593  halfnqq  7608  negsubdii  8442  halfpm6th  9342  decmul1  9652  i4  10876  fac4  10967  imi  11426  resqrexlemover  11536  ef01bndlem  12282  modsubi  12957  gcdmodi  12959  numexpp1  12962  karatsuba  12968  znnen  12984  sn0cld  14826  cospi  15489  sincos4thpi  15529  sincos3rdpi  15532  lgsdir2lem1  15722  lgsdir2lem5  15726  2lgsoddprmlem3d  15804  ex-bc  16148  ex-gcd  16150