Theorem eqtr2di 2257

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 2256	. 2 |- ( ph -> A = C )
4	3	eqcomd 2213	1 |- ( ph -> C = A )

Syntax hints:   -> wi 4   = wceq 1373

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 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200

This theorem is referenced by:  eqtr4id  2259  elpr2elpr  3830  elxp4  5189  elxp5  5190  fo1stresm  6270  fo2ndresm  6271  eloprabi  6305  fo2ndf  6336  xpsnen  6941  xpassen  6950  ac6sfi  7021  undifdc  7047  ine0  8501  nn0n0n1ge2  9478  fzval2  10168  fseq1p1m1  10251  fsum2dlemstep  11860  modfsummodlemstep  11883  fprod2dlemstep  12048  ef4p  12120  sin01bnd  12183  odd2np1  12299  sqpweven  12612  2sqpwodd  12613  psmetdmdm  14911  xmetdmdm  14943  dveflem  15313  reeff1oleme  15359  abssinper  15433  lgseisenlem1  15662