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  3901  dfdm4  4914  dmun  4929  resres  5016  inres  5021  resdifcom  5022  resiun1  5023  imainrect  5173  coundi  5229  coundir  5230  funopg  5351  offres  6278  mpomptsx  6341  cnvoprab  6378  snec  6741  halfpm6th  9327  numsucc  9613  decbin2  9714  fsumadd  11912  fsum2d  11941  fprodmul  12097  fprodfac  12121  fprodrec  12135  znnen  12964  txswaphmeolem  14988