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Theorem 3eqtr3a 2234
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, 3eqtrid 2222 . 2 (𝜑𝐴 = 𝐷)
51, 4eqtr3d 2212 1 (𝜑𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-cleq 2170
This theorem is referenced by:  uneqin  3388  coi2  5147  foima  5445  f1imacnv  5480  fvsnun2  5716  fnsnsplitdc  6508  phplem4  6857  phplem4on  6869  halfnqq  7411  resqrexlemcalc1  11025  absefib  11780  efieq1re  11781  restopnb  13720  cnmpt2t  13832  reeflog  14323  rpcxpsqrt  14381
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