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Theorem bnj602 31493
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 4845 . . 3 (𝑋 = 𝑌 → (𝑦𝑅𝑋𝑦𝑅𝑌))
21rabbidv 3371 . 2 (𝑋 = 𝑌 → {𝑦𝐴𝑦𝑅𝑋} = {𝑦𝐴𝑦𝑅𝑌})
3 df-bnj14 31266 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
4 df-bnj14 31266 . 2 pred(𝑌, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑌}
52, 3, 43eqtr4g 2856 1 (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  {crab 3091   class class class wbr 4841   predc-bnj14 31265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-bnj14 31266
This theorem is referenced by:  bnj601  31498  bnj852  31499  bnj18eq1  31505  bnj938  31515  bnj1125  31568  bnj1148  31572  bnj1318  31601  bnj1442  31625  bnj1450  31626  bnj1452  31628  bnj1463  31631  bnj1529  31646
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