Theorem 3eqtr3g 2285

Description: A chained equality inference, useful for converting from definitions. (Contributed by NM, 15-Nov-1994.)

Hypotheses

Ref	Expression
3eqtr3g.1	|- ( ph -> A = B )
3eqtr3g.2	|- A = C
3eqtr3g.3	|- B = D

Assertion

Ref	Expression
3eqtr3g	|- ( ph -> C = D )

Proof of Theorem 3eqtr3g

Step	Hyp	Ref	Expression
1		3eqtr3g.2	. . 3 |- A = C
2		3eqtr3g.1	. . 3 |- ( ph -> A = B )
3	1, 2	eqtr3id 2276	. 2 |- ( ph -> C = B )
4		3eqtr3g.3	. 2 |- B = D
5	3, 4	eqtrdi 2278	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1395

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  csbnest1g  3180  disjdif2  3570  dfopg  3854  xpid11  4946  sqxpeq0  5151  cores2  5240  funcoeqres  5602  dftpos2  6405  ine0  8536  fisumcom2  11944  fisum0diag2  11953  mertenslemi1  12041  fprodcom2fi  12132  fprodmodd  12147  bitsinv1  12468  4sqlem10  12905  setsslnid  13079  xpsff1o  13377  eqglact  13757  oppr1g  14040  dvmptccn  15383  dvmptc  15385  dvmptfsum  15393  fsumdvdsmul  15659  nninffeq  16345