Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  weeq2 Structured version   Visualization version   GIF version

Theorem weeq2 5063
 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 5045 . . 3 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 soeq2 5015 . . 3 (𝐴 = 𝐵 → (𝑅 Or 𝐴𝑅 Or 𝐵))
31, 2anbi12d 746 . 2 (𝐴 = 𝐵 → ((𝑅 Fr 𝐴𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐵𝑅 Or 𝐵)))
4 df-we 5035 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
5 df-we 5035 . 2 (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵𝑅 Or 𝐵))
63, 4, 53bitr4g 303 1 (𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   Or wor 4994   Fr wfr 5030   We wwe 5032 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-in 3562  df-ss 3569  df-po 4995  df-so 4996  df-fr 5033  df-we 5035 This theorem is referenced by:  ordeq  5689  0we1  7531  oieq2  8362  hartogslem1  8391  wemapwe  8538  ween  8802  dfac8  8901  weth  9261  fpwwe2cbv  9396  fpwwe2lem2  9398  fpwwe2lem5  9400  fpwwecbv  9410  fpwwelem  9411  canthwelem  9416  canthwe  9417  pwfseqlem4a  9427  pwfseqlem4  9428  ltweuz  12700  ltwenn  12701  bpolylem  14704  ltbwe  19391  vitali  23288  weeq12d  37090  aomclem6  37109  omeiunle  40038
 Copyright terms: Public domain W3C validator