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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 (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 2184 . 2 (𝜑𝐴 = 𝐷)
51, 4eqtr3d 2174 1 (𝜑𝐶 = 𝐷)
 Colors of variables: wff set class 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
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