Theorem 3eqtr3i 2117

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 2111	. 2 |- B = C
4		3eqtr3i.3	. 2 |- B = D
5	3, 4	eqtr3i 2111	1 |- C = D

Syntax hints:   = wceq 1290

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082

This theorem is referenced by:  csbvarg  2959  un12  3159  in12  3212  indif1  3245  difundir  3253  difindir  3255  dif32  3263  resmpt3  4774  xp0  4864  fvsnun1  5508  caov12  5847  caov13  5849  rec1nq  7015  halfnqq  7030  negsubdii  7828  halfpm6th  8697  decmul1  9001  i4  10118  fac4  10202  imi  10395  resqrexlemover  10504  ef01bndlem  11108  znnen  11550  sn0cld  11898  ex-bc  11929  ex-gcd  11931