MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  soeq1 Structured version   Visualization version   GIF version

Theorem soeq1 5567
Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.)
Assertion
Ref Expression
soeq1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))

Proof of Theorem soeq1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poeq1 5549 . . 3 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
2 breq 5108 . . . . 5 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 biidd 262 . . . . 5 (𝑅 = 𝑆 → (𝑥 = 𝑦𝑥 = 𝑦))
4 breq 5108 . . . . 5 (𝑅 = 𝑆 → (𝑦𝑅𝑥𝑦𝑆𝑥))
52, 3, 43orbi123d 1436 . . . 4 (𝑅 = 𝑆 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
652ralbidv 3213 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
71, 6anbi12d 632 . 2 (𝑅 = 𝑆 → ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥))))
8 df-so 5547 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
9 df-so 5547 . 2 (𝑆 Or 𝐴 ↔ (𝑆 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑆𝑦𝑥 = 𝑦𝑦𝑆𝑥)))
107, 8, 93bitr4g 314 1 (𝑅 = 𝑆 → (𝑅 Or 𝐴𝑆 Or 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3o 1087   = wceq 1542  wral 3065   class class class wbr 5106   Po wpo 5544   Or wor 5545
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 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-ex 1783  df-cleq 2729  df-clel 2815  df-ral 3066  df-br 5107  df-po 5546  df-so 5547
This theorem is referenced by:  weeq1  5622  ltsopi  10825  cnso  16130  opsrtoslem2  21466  soeq12d  41368
  Copyright terms: Public domain W3C validator