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  5278  elxp5  5279  fo1stresm  6399  fo2ndresm  6400  eloprabi  6436  fo2ndf  6467  xpsnen  7125  xpassen  7134  ac6sfi  7208  undifdc  7237  ine0  8728  nn0n0n1ge2  9724  fzval2  10430  fseq1p1m1  10517  hashfibclem  11303  hashf1  11308  fsum2dlemstep  12225  modfsummodlemstep  12248  fprod2dlemstep  12413  ef4p  12485  sin01bnd  12548  odd2np1  12664  sqpweven  12979  2sqpwodd  12980  psmetdmdm  15567  xmetdmdm  15599  dveflem  15969  reeff1oleme  16015  abssinper  16090  lgseisenlem1  16411