Theorem eqtr2i 2251

Description: An equality transitivity inference. (Contributed by NM, 21-Feb-1995.)

Hypotheses

Ref	Expression
eqtr2i.1	⊢ 𝐴 = 𝐵
eqtr2i.2	⊢ 𝐵 = 𝐶

Assertion

Ref	Expression
eqtr2i	⊢ 𝐶 = 𝐴

Proof of Theorem eqtr2i

Step	Hyp	Ref	Expression
1		eqtr2i.1	. . 3 ⊢ 𝐴 = 𝐵
2		eqtr2i.2	. . 3 ⊢ 𝐵 = 𝐶
3	1, 2	eqtri 2250	. 2 ⊢ 𝐴 = 𝐶
4	3	eqcomi 2233	1 ⊢ 𝐶 = 𝐴

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:  3eqtrri  2255  3eqtr2ri  2257  symdif1  3470  dfif3  3617  dfsn2  3681  prprc1  3778  ruv  4646  xpindi  4863  xpindir  4864  dmcnvcnv  4954  rncnvcnv  4955  imainrect  5180  dfrn4  5195  fcoi1  5514  foimacnv  5598  fsnunfv  5850  dfoprab3  6349  fiintim  7116  sbthlemi8  7154  pitonnlem1  8055  ixi  8753  recexaplem2  8822  zeo  9575  num0h  9612  dec10p  9643  fseq1p1m1  10319  cats1fvn  11335  fsumrelem  12022  ef0lem  12211  ef01bndlem  12307  3lcm2e6woprm  12648  strsl0  13121  0g0  13449  tgioo  15268  tgqioo  15269  dveflem  15440  sincos4thpi  15554  coskpi  15562