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Theorem weeq2 5666
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 5648 . . 3 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 soeq2 5611 . . 3 (𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))
31, 2anbi12d 632 . 2 (𝐴 = 𝐵 → ((𝑅 Fr 𝐴𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐵𝑅 Or 𝐵)))
4 df-we 5634 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
5 df-we 5634 . 2 (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵𝑅 Or 𝐵))
63, 4, 53bitr4g 314 1 (𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542   Or wor 5588   Fr wfr 5629   We wwe 5631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3956  df-ss 3966  df-po 5589  df-so 5590  df-fr 5632  df-we 5634
This theorem is referenced by:  ordeq  6372  0we1  8506  oieq2  9508  hartogslem1  9537  wemapwe  9692  ween  10030  dfac8  10130  weth  10490  fpwwe2cbv  10625  fpwwe2lem2  10627  fpwwe2lem4  10629  fpwwecbv  10639  fpwwelem  10640  canthwelem  10645  canthwe  10646  pwfseqlem4a  10656  pwfseqlem4  10657  ltweuz  13926  ltwenn  13927  bpolylem  15992  ltbwe  21599  vitali  25130  weeq12d  41782  aomclem6  41801  omeiunle  45233
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