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  6286  mpomptsx  6349  cnvoprab  6386  snec  6751  halfpm6th  9342  numsucc  9628  decbin2  9729  fsumadd  11932  fsum2d  11961  fprodmul  12117  fprodfac  12141  fprodrec  12155  znnen  12984  txswaphmeolem  15009