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Theorem 3eqtr3a 2174
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 (𝜑𝐴 = 𝐶)
3eqtr3a.3 (𝜑𝐵 = 𝐷)
Assertion
Ref Expression
3eqtr3a (𝜑𝐶 = 𝐷)

Proof of Theorem 3eqtr3a
StepHypRef Expression
1 3eqtr3a.2 . 2 (𝜑𝐴 = 𝐶)
2 3eqtr3a.1 . . 3 𝐴 = 𝐵
3 3eqtr3a.3 . . 3 (𝜑𝐵 = 𝐷)
42, 3syl5eq 2162 . 2 (𝜑𝐴 = 𝐷)
51, 4eqtr3d 2152 1 (𝜑𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316
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 1408  ax-gen 1410  ax-4 1472  ax-17 1491  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110
This theorem is referenced by:  uneqin  3297  coi2  5025  foima  5320  f1imacnv  5352  fvsnun2  5586  fnsnsplitdc  6369  phplem4  6717  phplem4on  6729  halfnqq  7186  resqrexlemcalc1  10754  absefib  11404  efieq1re  11405  restopnb  12277  cnmpt2t  12389
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