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  3775  ruv  4642  xpindi  4857  xpindir  4858  dmcnvcnv  4948  rncnvcnv  4949  imainrect  5174  dfrn4  5189  fcoi1  5508  foimacnv  5592  fsnunfv  5844  dfoprab3  6343  fiintim  7104  sbthlemi8  7142  pitonnlem1  8043  ixi  8741  recexaplem2  8810  zeo  9563  num0h  9600  dec10p  9631  fseq1p1m1  10302  cats1fvn  11311  fsumrelem  11997  ef0lem  12186  ef01bndlem  12282  3lcm2e6woprm  12623  strsl0  13096  0g0  13424  tgioo  15243  tgqioo  15244  dveflem  15415  sincos4thpi  15529  coskpi  15537