Theorem 3eqtr3i 2225

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 2219	. 2 ⊢ 𝐵 = 𝐶
4		3eqtr3i.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	eqtr3i 2219	1 ⊢ 𝐶 = 𝐷

Syntax hints:   = wceq 1364

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  csbvarg  3112  un12  3322  in12  3375  indif1  3409  difundir  3417  difindir  3419  dif32  3427  resmpt3  4996  xp0  5090  fvsnun1  5762  caov12  6116  caov13  6118  djuassen  7300  xpdjuen  7301  rec1nq  7479  halfnqq  7494  negsubdii  8328  halfpm6th  9228  decmul1  9537  i4  10751  fac4  10842  imi  11082  resqrexlemover  11192  ef01bndlem  11938  modsubi  12613  gcdmodi  12615  numexpp1  12618  karatsuba  12624  znnen  12640  sn0cld  14457  cospi  15120  sincos4thpi  15160  sincos3rdpi  15163  lgsdir2lem1  15353  lgsdir2lem5  15357  2lgsoddprmlem3d  15435  ex-bc  15459  ex-gcd  15461