Theorem 3eqtr3a 2244

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 2232	. 2 |- ( ph -> A = D )
5	1, 4	eqtr3d 2222	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1363

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 1457  ax-gen 1459  ax-4 1520  ax-17 1536  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-cleq 2180

This theorem is referenced by:  uneqin  3398  coi2  5157  foima  5455  f1imacnv  5490  fvsnun2  5727  fnsnsplitdc  6520  phplem4  6869  phplem4on  6881  halfnqq  7423  resqrexlemcalc1  11037  absefib  11792  efieq1re  11793  restopnb  14034  cnmpt2t  14146  reeflog  14637  rpcxpsqrt  14695