Theorem 3eqtr3g 2294

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 2285	. 2 |- ( ph -> C = B )
4		3eqtr3g.3	. 2 |- B = D
5	3, 4	eqtrdi 2287	1 |- ( ph -> C = D )

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:  csbnest1g  3203  disjdif2  3606  dfopg  3902  xpid11  5007  sqxpeq0  5214  cores2  5303  funcoeqres  5673  dftpos2  6536  ine0  8728  fisumcom2  12230  fisum0diag2  12239  mertenslemi1  12327  fprodcom2fi  12418  fprodmodd  12433  bitsinv1  12754  4sqlem10  13195  ballotfilemgun  13326  setsslnid  13462  xpsff1o  13729  eqglact  14087  gsum2dfilem2  14272  oppr1g  14497  dvmptccn  15941  dvmptc  15943  dvmptfsum  15951  chtdif  16264  ppidif  16269  fsumdvdsmul  16285  nninffeq  17269