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Theorem 3eqtrrd 2077
 Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
3eqtrd.1 (𝜑𝐴 = 𝐵)
3eqtrd.2 (𝜑𝐵 = 𝐶)
3eqtrd.3 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
3eqtrrd (𝜑𝐷 = 𝐴)

Proof of Theorem 3eqtrrd
StepHypRef Expression
1 3eqtrd.1 . . 3 (𝜑𝐴 = 𝐵)
2 3eqtrd.2 . . 3 (𝜑𝐵 = 𝐶)
31, 2eqtrd 2072 . 2 (𝜑𝐴 = 𝐶)
4 3eqtrd.3 . 2 (𝜑𝐶 = 𝐷)
53, 4eqtr2d 2073 1 (𝜑𝐷 = 𝐴)
 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:  nnanq0  6556  1idprl  6688  1idpru  6689  axcnre  6955  fseq1p1m1  8956  expmulzap  9301  expubnd  9311  subsq  9358  crim  9458  rereb  9463
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