Theorem 3eqtr3g 2143

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	syl5eqr 2134	. 2 |- ( ph -> C = B )
4		3eqtr3g.3	. 2 |- B = D
5	3, 4	syl6eq 2136	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-cleq 2081

This theorem is referenced by:  csbnest1g  2983  disjdif2  3361  dfopg  3620  cores2  4943  funcoeqres  5284  dftpos2  6026  ine0  7870  fisumcom2  10828  fisum0diag2  10837  mertenslemi1  10925  setsnidn  11540