Theorem 3eqtr3a 2196

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 2184	. 2 |- ( ph -> A = D )
5	1, 4	eqtr3d 2174	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132

This theorem is referenced by:  uneqin  3327  coi2  5055  foima  5350  f1imacnv  5384  fvsnun2  5618  fnsnsplitdc  6401  phplem4  6749  phplem4on  6761  halfnqq  7218  resqrexlemcalc1  10786  absefib  11477  efieq1re  11478  restopnb  12350  cnmpt2t  12462