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Theorem 3eqtrrd 2093
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 2088 . 2 (𝜑𝐴 = 𝐶)
4 3eqtrd.3 . 2 (𝜑𝐶 = 𝐷)
53, 4eqtr2d 2089 1 (𝜑𝐷 = 𝐴)
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:  nnanq0  6613  1idprl  6745  1idpru  6746  axcnre  7012  fseq1p1m1  9057  expmulzap  9465  expubnd  9476  subsq  9524  bcm1k  9627  bcpasc  9633  crim  9685  rereb  9690
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