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Theorem bnj602 31759
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 4960 . . 3 (𝑋 = 𝑌 → (𝑦𝑅𝑋𝑦𝑅𝑌))
21rabbidv 3420 . 2 (𝑋 = 𝑌 → {𝑦𝐴𝑦𝑅𝑋} = {𝑦𝐴𝑦𝑅𝑌})
3 df-bnj14 31532 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
4 df-bnj14 31532 . 2 pred(𝑌, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑌}
52, 3, 43eqtr4g 2854 1 (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1520  {crab 3107   class class class wbr 4956   predc-bnj14 31531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-br 4957  df-bnj14 31532
This theorem is referenced by:  bnj601  31764  bnj852  31765  bnj18eq1  31771  bnj938  31781  bnj1125  31834  bnj1148  31838  bnj1318  31867  bnj1442  31891  bnj1450  31892  bnj1452  31894  bnj1463  31897  bnj1529  31912
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