Theorem 3eqtr3g 2233

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 2224	. 2 |- ( ph -> C = B )
4		3eqtr3g.3	. 2 |- B = D
5	3, 4	eqtrdi 2226	1 |- ( ph -> C = D )

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:  csbnest1g  3112  disjdif2  3501  dfopg  3775  xpid11  4847  sqxpeq0  5049  cores2  5138  funcoeqres  5489  dftpos2  6257  ine0  8345  fisumcom2  11437  fisum0diag2  11446  mertenslemi1  11534  fprodcom2fi  11625  fprodmodd  11640  4sqlem10  12375  setsslnid  12504  oppr1g  13151  dvmptccn  13961  nninffeq  14540