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Theorem weeq2 5301
Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
weeq2 (𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))

Proof of Theorem weeq2
StepHypRef Expression
1 freq2 5283 . . 3 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 soeq2 5253 . . 3 (𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))
31, 2anbi12d 625 . 2 (𝐴 = 𝐵 → ((𝑅 Fr 𝐴𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐵𝑅 Or 𝐵)))
4 df-we 5273 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
5 df-we 5273 . 2 (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵𝑅 Or 𝐵))
63, 4, 53bitr4g 306 1 (𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653   Or wor 5232   Fr wfr 5268   We wwe 5270
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-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-ral 3094  df-in 3776  df-ss 3783  df-po 5233  df-so 5234  df-fr 5271  df-we 5273
This theorem is referenced by:  ordeq  5948  0we1  7826  oieq2  8660  hartogslem1  8689  wemapwe  8844  ween  9144  dfac8  9245  weth  9605  fpwwe2cbv  9740  fpwwe2lem2  9742  fpwwe2lem5  9744  fpwwecbv  9754  fpwwelem  9755  canthwelem  9760  canthwe  9761  pwfseqlem4a  9771  pwfseqlem4  9772  ltweuz  13015  ltwenn  13016  bpolylem  15115  ltbwe  19795  vitali  23721  weeq12d  38395  aomclem6  38414  omeiunle  41477
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