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Theorem bnj602 33567
Description: Equality theorem for the pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Assertion
Ref Expression
bnj602 (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))

Proof of Theorem bnj602
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 breq2 5114 . . 3 (𝑋 = 𝑌 → (𝑦𝑅𝑋𝑦𝑅𝑌))
21rabbidv 3418 . 2 (𝑋 = 𝑌 → {𝑦𝐴𝑦𝑅𝑋} = {𝑦𝐴𝑦𝑅𝑌})
3 df-bnj14 33341 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
4 df-bnj14 33341 . 2 pred(𝑌, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑌}
52, 3, 43eqtr4g 2802 1 (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  {crab 3410   class class class wbr 5110   predc-bnj14 33340
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 2708
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 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-bnj14 33341
This theorem is referenced by:  bnj601  33572  bnj852  33573  bnj18eq1  33579  bnj938  33589  bnj1125  33644  bnj1148  33648  bnj1318  33677  bnj1442  33701  bnj1450  33702  bnj1452  33704  bnj1463  33707  bnj1529  33722
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