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Theorem bnj602 32427
 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 5040 . . 3 (𝑋 = 𝑌 → (𝑦𝑅𝑋𝑦𝑅𝑌))
21rabbidv 3392 . 2 (𝑋 = 𝑌 → {𝑦𝐴𝑦𝑅𝑋} = {𝑦𝐴𝑦𝑅𝑌})
3 df-bnj14 32199 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
4 df-bnj14 32199 . 2 pred(𝑌, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑌}
52, 3, 43eqtr4g 2818 1 (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  {crab 3074   class class class wbr 5036   predc-bnj14 32198 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-un 3865  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-bnj14 32199 This theorem is referenced by:  bnj601  32432  bnj852  32433  bnj18eq1  32439  bnj938  32449  bnj1125  32504  bnj1148  32508  bnj1318  32537  bnj1442  32561  bnj1450  32562  bnj1452  32564  bnj1463  32567  bnj1529  32582
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