Theorem 3eqtr3a 2138

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 2126	. 2 |- ( ph -> A = D )
5	1, 4	eqtr3d 2116	1 |- ( ph -> C = D )

Syntax hints:   -> wi 4   = wceq 1285

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-cleq 2075

This theorem is referenced by:  uneqin  3222  coi2  4867  foima  5142  f1imacnv  5174  fvsnun2  5393  phplem4  6390  phplem4on  6402  halfnqq  6662  resqrexlemcalc1  10038