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

Theorem weeq2 5673
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 5653 . . 3 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 soeq2 5614 . . 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 5639 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
5 df-we 5639 . 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 206  wa 395   = wceq 1540   Or wor 5591   Fr wfr 5634   We wwe 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-ral 3062  df-ss 3968  df-po 5592  df-so 5593  df-fr 5637  df-we 5639
This theorem is referenced by:  weeq12d  5674  ordeq  6391  0we1  8544  oieq2  9553  wemapwe  9737  ween  10075  dfac8  10176  weth  10535  pwfseqlem4a  10701  pwfseqlem4  10702  ltweuz  14002  ltwenn  14003  bpolylem  16084  ltbwe  22062  vitali  25648  aomclem6  43071  omeiunle  46532
  Copyright terms: Public domain W3C validator