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Theorem eqbrtrrdi 5188
Description: A chained equality inference for a binary relation. (Contributed by NM, 4-Jan-2006.)
Hypotheses
Ref Expression
eqbrtrrdi.1 (𝜑𝐵 = 𝐴)
eqbrtrrdi.2 𝐵𝑅𝐶
Assertion
Ref Expression
eqbrtrrdi (𝜑𝐴𝑅𝐶)

Proof of Theorem eqbrtrrdi
StepHypRef Expression
1 eqbrtrrdi.1 . . 3 (𝜑𝐵 = 𝐴)
21eqcomd 2741 . 2 (𝜑𝐴 = 𝐵)
3 eqbrtrrdi.2 . 2 𝐵𝑅𝐶
42, 3eqbrtrdi 5187 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537   class class class wbr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149
This theorem is referenced by:  grur1  10858  t1connperf  23460  basellem9  27147  sqff1o  27240  ballotlemic  34488  ballotlem1c  34489  pibt2  37400
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