Theorem 3eqtr4ri 2171

Description: An inference from three chained equalities. (Contributed by NM, 2-Sep-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
3eqtr4i.1	|- A = B
3eqtr4i.2	|- C = A
3eqtr4i.3	|- D = B

Assertion

Ref	Expression
3eqtr4ri	|- D = C

Proof of Theorem 3eqtr4ri

Step	Hyp	Ref	Expression
1		3eqtr4i.3	. . 3 |- D = B
2		3eqtr4i.1	. . 3 |- A = B
3	1, 2	eqtr4i 2163	. 2 |- D = A
4		3eqtr4i.2	. 2 |- C = A
5	3, 4	eqtr4i 2163	1 |- D = C

Syntax hints:   = wceq 1331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132

This theorem is referenced by:  cbvreucsf  3064  dfif6  3476  qdass  3620  tpidm12  3622  unipr  3750  dfdm4  4731  dmun  4746  resres  4831  inres  4836  resdifcom  4837  resiun1  4838  imainrect  4984  coundi  5040  coundir  5041  funopg  5157  offres  6033  mpomptsx  6095  cnvoprab  6131  snec  6490  halfpm6th  8940  numsucc  9221  decbin2  9322  fsumadd  11175  fsum2d  11204  znnen  11911