Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  poeq1 Structured version   Unicode version

Theorem poeq1 4506
 Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
poeq1

Proof of Theorem poeq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4214 . . . . . 6
21notbid 286 . . . . 5
3 breq 4214 . . . . . . 7
4 breq 4214 . . . . . . 7
53, 4anbi12d 692 . . . . . 6
6 breq 4214 . . . . . 6
75, 6imbi12d 312 . . . . 5
82, 7anbi12d 692 . . . 4
98ralbidv 2725 . . 3
1092ralbidv 2747 . 2
11 df-po 4503 . 2
12 df-po 4503 . 2
1310, 11, 123bitr4g 280 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   wceq 1652  wral 2705   class class class wbr 4212   wpo 4501 This theorem is referenced by:  soeq1  4522 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-cleq 2429  df-clel 2432  df-ral 2710  df-br 4213  df-po 4503
 Copyright terms: Public domain W3C validator