Theorem 3eqtr3a 2151

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	syl5eq 2139	. 2 |- ( ph -> A = D )
5	1, 4	eqtr3d 2129	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1296

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-4 1452  ax-17 1471  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-cleq 2088

This theorem is referenced by:  uneqin  3266  coi2  4981  foima  5273  f1imacnv  5305  fvsnun2  5534  fnsnsplitdc  6304  phplem4  6651  phplem4on  6663  halfnqq  7066  resqrexlemcalc1  10578  absefib  11224  efieq1re  11225  restopnb  12048