Theorem eqtr2di 2246

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 2245	. 2 |- ( ph -> A = C )
4	3	eqcomd 2202	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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  eqtr4id  2248  elxp4  5157  elxp5  5158  fo1stresm  6219  fo2ndresm  6220  eloprabi  6254  fo2ndf  6285  xpsnen  6880  xpassen  6889  ac6sfi  6959  undifdc  6985  ine0  8420  nn0n0n1ge2  9396  fzval2  10086  fseq1p1m1  10169  fsum2dlemstep  11599  modfsummodlemstep  11622  fprod2dlemstep  11787  ef4p  11859  sin01bnd  11922  odd2np1  12038  sqpweven  12343  2sqpwodd  12344  psmetdmdm  14560  xmetdmdm  14592  dveflem  14962  reeff1oleme  15008  abssinper  15082  lgseisenlem1  15311