Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj602 Structured version   Visualization version   GIF version

Theorem bnj602 30959
 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 4648 . . 3 (𝑋 = 𝑌 → (𝑦𝑅𝑋𝑦𝑅𝑌))
21rabbidv 3184 . 2 (𝑋 = 𝑌 → {𝑦𝐴𝑦𝑅𝑋} = {𝑦𝐴𝑦𝑅𝑌})
3 df-bnj14 30729 . 2 pred(𝑋, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑋}
4 df-bnj14 30729 . 2 pred(𝑌, 𝐴, 𝑅) = {𝑦𝐴𝑦𝑅𝑌}
52, 3, 43eqtr4g 2679 1 (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1481  {crab 2913   class class class wbr 4644   predc-bnj14 30728 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-bnj14 30729 This theorem is referenced by:  bnj601  30964  bnj852  30965  bnj18eq1  30971  bnj938  30981  bnj1125  31034  bnj1148  31038  bnj1318  31067  bnj1442  31091  bnj1450  31092  bnj1452  31094  bnj1463  31097  bnj1529  31112
 Copyright terms: Public domain W3C validator