Theorem 3eqtr4ri 2169

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 2161	. 2 ⊢ 𝐷 = 𝐴
4		3eqtr4i.2	. 2 ⊢ 𝐶 = 𝐴
5	3, 4	eqtr4i 2161	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 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130

This theorem is referenced by:  cbvreucsf  3059  dfif6  3471  qdass  3615  tpidm12  3617  unipr  3745  dfdm4  4726  dmun  4741  resres  4826  inres  4831  resdifcom  4832  resiun1  4833  imainrect  4979  coundi  5035  coundir  5036  funopg  5152  offres  6026  mpomptsx  6088  cnvoprab  6124  snec  6483  halfpm6th  8933  numsucc  9214  decbin2  9315  fsumadd  11168  fsum2d  11197  znnen  11900