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  3469  dfif3  3616  dfsn2  3680  prprc1  3774  ruv  4641  xpindi  4856  xpindir  4857  dmcnvcnv  4947  rncnvcnv  4948  imainrect  5173  dfrn4  5188  fcoi1  5505  foimacnv  5589  fsnunfv  5839  dfoprab3  6335  fiintim  7089  sbthlemi8  7127  pitonnlem1  8028  ixi  8726  recexaplem2  8795  zeo  9548  num0h  9585  dec10p  9616  fseq1p1m1  10286  cats1fvn  11291  fsumrelem  11977  ef0lem  12166  ef01bndlem  12262  3lcm2e6woprm  12603  strsl0  13076  0g0  13404  tgioo  15222  tgqioo  15223  dveflem  15394  sincos4thpi  15508  coskpi  15516