Theorem eqtr2di 2288

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 2287	. 2 |- ( ph -> A = C )
4	3	eqcomd 2244	1 |- ( ph -> C = A )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  eqtr4id  2290  elpr2elpr  3901  elxp4  5275  elxp5  5276  fo1stresm  6395  fo2ndresm  6396  eloprabi  6432  fo2ndf  6463  xpsnen  7119  xpassen  7128  ac6sfi  7202  undifdc  7231  ine0  8723  nn0n0n1ge2  9720  fzval2  10425  fseq1p1m1  10512  hashfibclem  11298  hashf1  11303  fsum2dlemstep  12220  modfsummodlemstep  12243  fprod2dlemstep  12408  ef4p  12480  sin01bnd  12543  odd2np1  12659  sqpweven  12974  2sqpwodd  12975  psmetdmdm  15516  xmetdmdm  15548  dveflem  15918  reeff1oleme  15964  abssinper  16039  lgseisenlem1  16355