Theorem 3eqtr3a 2286

Description: A chained equality inference, useful for converting from definitions. (Contributed by Mario Carneiro, 6-Nov-2015.)

Hypotheses

Ref	Expression
3eqtr3a.1	|- A = B
3eqtr3a.2	|- ( ph -> A = C )
3eqtr3a.3	|- ( ph -> B = D )

Assertion

Ref	Expression
3eqtr3a	|- ( ph -> C = D )

Proof of Theorem 3eqtr3a

Step	Hyp	Ref	Expression
1		3eqtr3a.2	. 2 |- ( ph -> A = C )
2		3eqtr3a.1	. . 3 |- A = B
3		3eqtr3a.3	. . 3 |- ( ph -> B = D )
4	2, 3	eqtrid 2274	. 2 |- ( ph -> A = D )
5	1, 4	eqtr3d 2264	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1395

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  uneqin  3455  coi2  5245  foima  5553  f1imacnv  5589  fvsnun2  5837  fnsnsplitdc  6651  phplem4  7016  phplem4on  7029  halfnqq  7597  resqrexlemcalc1  11525  absefib  12282  efieq1re  12283  restopnb  14855  cnmpt2t  14967  reeflog  15537  rpcxpsqrt  15596