Theorem eqtr2di 2281

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 2280	. 2 |- ( ph -> A = C )
4	3	eqcomd 2237	1 |- ( ph -> C = A )

Syntax hints:   -> wi 4   = wceq 1398

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  eqtr4id  2283  elpr2elpr  3864  elxp4  5231  elxp5  5232  fo1stresm  6333  fo2ndresm  6334  eloprabi  6370  fo2ndf  6401  xpsnen  7048  xpassen  7057  ac6sfi  7130  undifdc  7159  ine0  8632  nn0n0n1ge2  9611  fzval2  10308  fseq1p1m1  10391  fsum2dlemstep  12075  modfsummodlemstep  12098  fprod2dlemstep  12263  ef4p  12335  sin01bnd  12398  odd2np1  12514  sqpweven  12827  2sqpwodd  12828  psmetdmdm  15135  xmetdmdm  15167  dveflem  15537  reeff1oleme  15583  abssinper  15657  lgseisenlem1  15889