Theorem eqtr2di 2279

Description: An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Hypotheses

Ref	Expression
eqtr2di.1	|- ( ph -> A = B )
eqtr2di.2	|- B = C

Assertion

Ref	Expression
eqtr2di	|- ( ph -> C = A )

Proof of Theorem eqtr2di

Step	Hyp	Ref	Expression
1		eqtr2di.1	. . 3 |- ( ph -> A = B )
2		eqtr2di.2	. . 3 |- B = C
3	1, 2	eqtrdi 2278	. 2 |- ( ph -> A = C )
4	3	eqcomd 2235	1 |- ( ph -> C = A )

Syntax hints:   -> wi 4   = 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:  eqtr4id  2281  elpr2elpr  3854  elxp4  5216  elxp5  5217  fo1stresm  6307  fo2ndresm  6308  eloprabi  6342  fo2ndf  6373  xpsnen  6980  xpassen  6989  ac6sfi  7060  undifdc  7086  ine0  8540  nn0n0n1ge2  9517  fzval2  10207  fseq1p1m1  10290  fsum2dlemstep  11945  modfsummodlemstep  11968  fprod2dlemstep  12133  ef4p  12205  sin01bnd  12268  odd2np1  12384  sqpweven  12697  2sqpwodd  12698  psmetdmdm  14998  xmetdmdm  15030  dveflem  15400  reeff1oleme  15446  abssinper  15520  lgseisenlem1  15749