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Theorem bnj1152 30827
 Description: Technical lemma for bnj69 30839. 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 4626 . 2 (𝑦 = 𝑌 → (𝑦𝑅𝑋𝑌𝑅𝑋))
2 df-bnj14 30515 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
31, 2elrab2 3353 1 (𝑌 ∈ pred(𝑋, 𝐴, 𝑅) ↔ (𝑌𝐴𝑌𝑅𝑋))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1987   class class class wbr 4623   predc-bnj14 30514 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-bnj14 30515 This theorem is referenced by:  bnj1175  30833  bnj1177  30835  bnj1388  30862
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