Theorem 3eqtr3g 2195

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 2186	. 2 |- ( ph -> C = B )
4		3eqtr3g.3	. 2 |- B = D
5	3, 4	syl6eq 2188	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132

This theorem is referenced by:  csbnest1g  3055  disjdif2  3441  dfopg  3703  xpid11  4762  sqxpeq0  4962  cores2  5051  funcoeqres  5398  dftpos2  6158  ine0  8156  fisumcom2  11207  fisum0diag2  11216  mertenslemi1  11304  setsslnid  12010  dvmptccn  12848  nninffeq  13216