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  6297  mpomptsx  6362  cnvoprab  6399  snec  6765  halfpm6th  9364  numsucc  9650  decbin2  9751  fsumadd  11972  fsum2d  12001  fprodmul  12157  fprodfac  12181  fprodrec  12195  znnen  13024  txswaphmeolem  15050