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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
StepHypRef Expression
1 3eqtr3a.2 . 2  |-  ( ph  ->  A  =  C )
2 3eqtr3a.1 . . 3  |-  A  =  B
3 3eqtr3a.3 . . 3  |-  ( ph  ->  B  =  D )
42, 3syl5eq 2126 . 2  |-  ( ph  ->  A  =  D )
51, 4eqtr3d 2116 1  |-  ( ph  ->  C  =  D )
Colors of variables: wff set class
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
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