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Theorem 3eqtr3a 2214
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 2202 . 2 (𝜑𝐴 = 𝐷)
51, 4eqtr3d 2192 1 (𝜑𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335
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 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-cleq 2150
This theorem is referenced by:  uneqin  3359  coi2  5105  foima  5400  f1imacnv  5434  fvsnun2  5668  fnsnsplitdc  6455  phplem4  6803  phplem4on  6815  halfnqq  7333  resqrexlemcalc1  10926  absefib  11679  efieq1re  11680  restopnb  12677  cnmpt2t  12789  reeflog  13280  rpcxpsqrt  13338
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