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Theorem 3eqtr3a 2112
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 2100 . 2 (𝜑𝐴 = 𝐷)
51, 4eqtr3d 2090 1 (𝜑𝐶 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049
This theorem is referenced by:  uneqin  3216  coi2  4865  foima  5139  f1imacnv  5171  fvsnun2  5389  phplem4  6349  phplem4on  6360  halfnqq  6566  resqrexlemcalc1  9841
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