Theorem 3eqtr3i 2260

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 2254	. 2 ⊢ 𝐵 = 𝐶
4		3eqtr3i.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	eqtr3i 2254	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 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  csbvarg  3156  un12  3367  in12  3420  indif1  3454  difundir  3462  difindir  3464  dif32  3472  resmpt3  5068  xp0  5163  fvsnun1  5859  caov12  6221  caov13  6223  djuassen  7475  xpdjuen  7476  rec1nq  7658  halfnqq  7673  negsubdii  8507  halfpm6th  9407  decmul1  9719  i4  10950  fac4  11041  imi  11523  resqrexlemover  11633  ef01bndlem  12380  modsubi  13055  gcdmodi  13057  numexpp1  13060  karatsuba  13066  znnen  13082  sn0cld  14931  cospi  15594  sincos4thpi  15634  sincos3rdpi  15637  lgsdir2lem1  15830  lgsdir2lem5  15834  2lgsoddprmlem3d  15912  ex-bc  16426  ex-gcd  16428