Theorem eqtr2di 2243

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 2242	. 2 |- ( ph -> A = C )
4	3	eqcomd 2199	1 |- ( ph -> C = A )

Syntax hints:   -> wi 4   = wceq 1364

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186

This theorem is referenced by:  eqtr4id  2245  elxp4  5154  elxp5  5155  fo1stresm  6216  fo2ndresm  6217  eloprabi  6251  fo2ndf  6282  xpsnen  6877  xpassen  6886  ac6sfi  6956  undifdc  6982  ine0  8415  nn0n0n1ge2  9390  fzval2  10080  fseq1p1m1  10163  fsum2dlemstep  11580  modfsummodlemstep  11603  fprod2dlemstep  11768  ef4p  11840  sin01bnd  11903  odd2np1  12017  sqpweven  12316  2sqpwodd  12317  psmetdmdm  14503  xmetdmdm  14535  dveflem  14905  reeff1oleme  14948  abssinper  15022  lgseisenlem1  15227