Theorem 3eqtr2rd 2217

Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.)

Hypotheses

Ref	Expression
3eqtr2d.1	|- ( ph -> A = B )
3eqtr2d.2	|- ( ph -> C = B )
3eqtr2d.3	|- ( ph -> C = D )

Assertion

Ref	Expression
3eqtr2rd	|- ( ph -> D = A )

Proof of Theorem 3eqtr2rd

Step	Hyp	Ref	Expression
1		3eqtr2d.1	. . 3 |- ( ph -> A = B )
2		3eqtr2d.2	. . 3 |- ( ph -> C = B )
3	1, 2	eqtr4d 2213	. 2 |- ( ph -> A = C )
4		3eqtr2d.3	. 2 |- ( ph -> C = D )
5	3, 4	eqtr2d 2211	1 |- ( ph -> D = A )

Syntax hints:   -> wi 4   = wceq 1353

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  difinfsn  7099  nnnninfeq  7126  prarloclemlo  7493  recexgt0sr  7772  xp1d2m1eqxm1d2  9171  qnegmod  10369  modqeqmodmin  10394  faclbnd2  10722  cjmulval  10897  fsumsplit  11415  fzosump1  11425  isumclim3  11431  bcxmas  11497  trireciplem  11508  geo2sum  11522  geo2lim  11524  geoisum1c  11528  cvgratnnlemseq  11534  mertenslemi1  11543  fprodsplitdc  11604  eftlub  11698  addsin  11750  subsin  11751  subcos  11755  qredeu  12097  nn0sqrtelqelz  12206  strslfv2d  12505  mulgaddcomlem  13006  dvexp  14178  tangtx  14262  logsqrt  14346  2sqlem8  14473