Theorem eqtr2i 2218

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 2217	. 2 ⊢ 𝐴 = 𝐶
4	3	eqcomi 2200	1 ⊢ 𝐶 = 𝐴

Syntax hints:   = wceq 1364

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  3eqtrri  2222  3eqtr2ri  2224  symdif1  3428  dfif3  3574  dfsn2  3636  prprc1  3730  ruv  4586  xpindi  4801  xpindir  4802  dmcnvcnv  4890  rncnvcnv  4891  imainrect  5115  dfrn4  5130  fcoi1  5438  foimacnv  5522  fsnunfv  5763  dfoprab3  6249  fiintim  6992  sbthlemi8  7030  pitonnlem1  7912  ixi  8610  recexaplem2  8679  zeo  9431  num0h  9468  dec10p  9499  fseq1p1m1  10169  fsumrelem  11636  ef0lem  11825  ef01bndlem  11921  3lcm2e6woprm  12254  strsl0  12727  0g0  13019  tgioo  14790  tgqioo  14791  dveflem  14962  sincos4thpi  15076  coskpi  15084