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Theorem eqbrtrrdi 5182
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 2742 . 2 (𝜑𝐴 = 𝐵)
3 eqbrtrrdi.2 . 2 𝐵𝑅𝐶
42, 3eqbrtrdi 5181 1 (𝜑𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539   class class class wbr 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143
This theorem is referenced by:  grur1  10861  t1connperf  23445  basellem9  27133  sqff1o  27226  ballotlemic  34510  ballotlem1c  34511  pibt2  37419
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