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Theorem bnj1152 33667
Description: Technical lemma for bnj69 33679. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj1152 (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌𝐴𝑌𝑅𝑋))

Proof of Theorem bnj1152
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 breq1 5109 . 2 (𝑦 = 𝑌 → (𝑦𝑅𝑋𝑌𝑅𝑋))
2 df-bnj14 33358 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
31, 2elrab2 3649 1 (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌𝐴𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107   class class class wbr 5106   predc-bnj14 33357
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 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-bnj14 33358
This theorem is referenced by:  bnj1175  33673  bnj1177  33675  bnj1388  33702
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