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Theorem eqbrtrrdi 5187
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 2739 . 2 (𝜑𝐴 = 𝐵)
3 eqbrtrrdi.2 . 2 𝐵𝑅𝐶
42, 3eqbrtrdi 5186 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542   class class class wbr 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148
This theorem is referenced by:  grur1  10811  t1connperf  22922  basellem9  26573  sqff1o  26666  ballotlemic  33443  ballotlem1c  33444  pibt2  36236
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