Theorem 3eqtr4ri 2263

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 2255	. 2 ⊢ 𝐷 = 𝐴
4		3eqtr4i.2	. 2 ⊢ 𝐶 = 𝐴
5	3, 4	eqtr4i 2255	1 ⊢ 𝐷 = 𝐶

Syntax hints:   = wceq 1397

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2213

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

This theorem is referenced by:  cbvreucsf  3192  dfif6  3607  qdass  3768  tpidm12  3770  unipr  3907  dfdm4  4923  dmun  4938  resres  5025  inres  5030  resdifcom  5031  resiun1  5032  imainrect  5182  coundi  5238  coundir  5239  funopg  5360  offres  6296  mpomptsx  6361  cnvoprab  6398  snec  6764  halfpm6th  9363  numsucc  9649  decbin2  9750  fsumadd  11966  fsum2d  11995  fprodmul  12151  fprodfac  12175  fprodrec  12189  znnen  13018  txswaphmeolem  15043