Theorem 3eqtr4ri 2261

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 2253	. 2 |- D = A
4		3eqtr4i.2	. 2 |- C = A
5	3, 4	eqtr4i 2253	1 |- D = C

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:  cbvreucsf  3189  dfif6  3604  qdass  3763  tpidm12  3765  unipr  3902  dfdm4  4915  dmun  4930  resres  5017  inres  5022  resdifcom  5023  resiun1  5024  imainrect  5174  coundi  5230  coundir  5231  funopg  5352  offres  6280  mpomptsx  6343  cnvoprab  6380  snec  6743  halfpm6th  9331  numsucc  9617  decbin2  9718  fsumadd  11917  fsum2d  11946  fprodmul  12102  fprodfac  12126  fprodrec  12140  znnen  12969  txswaphmeolem  14994