Theorem 3eqtr3g 2249

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 2240	. 2 |- ( ph -> C = B )
4		3eqtr3g.3	. 2 |- B = D
5	3, 4	eqtrdi 2242	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1364

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-cleq 2186

This theorem is referenced by:  csbnest1g  3136  disjdif2  3525  dfopg  3802  xpid11  4885  sqxpeq0  5089  cores2  5178  funcoeqres  5531  dftpos2  6314  ine0  8413  fisumcom2  11581  fisum0diag2  11590  mertenslemi1  11678  fprodcom2fi  11769  fprodmodd  11784  4sqlem10  12525  setsslnid  12670  xpsff1o  12932  eqglact  13295  oppr1g  13578  dvmptccn  14864  nninffeq  15510