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Theorem 3eqtrrd 2074
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 2069 . 2 (φA = 𝐶)
4 3eqtrd.3 . 2 (φ𝐶 = 𝐷)
53, 4eqtr2d 2070 1 (φ𝐷 = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242
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 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030
This theorem is referenced by:  nnanq0  6440  1idprl  6564  1idpru  6565  axcnre  6725  fseq1p1m1  8686  expmulzap  8915  expubnd  8925  subsq  8971  crim  9046  rereb  9051
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