Theorem 3eqtr3a 2250

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 2238	. 2 |- ( ph -> A = D )
5	1, 4	eqtr3d 2228	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:  uneqin  3411  coi2  5183  foima  5482  f1imacnv  5518  fvsnun2  5757  fnsnsplitdc  6560  phplem4  6913  phplem4on  6925  halfnqq  7472  resqrexlemcalc1  11161  absefib  11917  efieq1re  11918  restopnb  14360  cnmpt2t  14472  reeflog  15039  rpcxpsqrt  15097