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Theorem 3eqtr4a 2098
 Description: A chained equality inference, useful for converting to definitions. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
3eqtr4a.1 𝐴 = 𝐵
3eqtr4a.2 (𝜑𝐶 = 𝐴)
3eqtr4a.3 (𝜑𝐷 = 𝐵)
Assertion
Ref Expression
3eqtr4a (𝜑𝐶 = 𝐷)

Proof of Theorem 3eqtr4a
StepHypRef Expression
1 3eqtr4a.2 . . 3 (𝜑𝐶 = 𝐴)
2 3eqtr4a.1 . . 3 𝐴 = 𝐵
31, 2syl6eq 2088 . 2 (𝜑𝐶 = 𝐵)
4 3eqtr4a.3 . 2 (𝜑𝐷 = 𝐵)
53, 4eqtr4d 2075 1 (𝜑𝐶 = 𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033 This theorem is referenced by:  uniintsnr  3651  fndmdifcom  5273  offres  5762  1stval2  5782  2ndval2  5783  ecovcom  6213  ecovass  6215  ecovdi  6217  zeo  8343  xnegneg  8746  fzsuc2  8941  expnegap0  9263  absexp  9675  sqr2irrlem  9877
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