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Theorem bnj1152 32167
Description: Technical lemma for bnj69 32179. 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 5060 . 2 (𝑦 = 𝑌 → (𝑦𝑅𝑋𝑌𝑅𝑋))
2 df-bnj14 31858 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
31, 2elrab2 3680 1 (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌𝐴𝑌𝑅𝑋))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105   class class class wbr 5057   predc-bnj14 31857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-bnj14 31858
This theorem is referenced by:  bnj1175  32173  bnj1177  32175  bnj1388  32202
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