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Theorem poeq1 5236
Description: Equality theorem for partial ordering predicate. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
poeq1 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))

Proof of Theorem poeq1
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 4845 . . . . . 6 (𝑅 = 𝑆 → (𝑥𝑅𝑥𝑥𝑆𝑥))
21notbid 310 . . . . 5 (𝑅 = 𝑆 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑆𝑥))
3 breq 4845 . . . . . . 7 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
4 breq 4845 . . . . . . 7 (𝑅 = 𝑆 → (𝑦𝑅𝑧𝑦𝑆𝑧))
53, 4anbi12d 625 . . . . . 6 (𝑅 = 𝑆 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑆𝑦𝑦𝑆𝑧)))
6 breq 4845 . . . . . 6 (𝑅 = 𝑆 → (𝑥𝑅𝑧𝑥𝑆𝑧))
75, 6imbi12d 336 . . . . 5 (𝑅 = 𝑆 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
82, 7anbi12d 625 . . . 4 (𝑅 = 𝑆 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑆𝑥 ∧ ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧))))
98ralbidv 3167 . . 3 (𝑅 = 𝑆 → (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑧𝐴𝑥𝑆𝑥 ∧ ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧))))
1092ralbidv 3170 . 2 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑆𝑥 ∧ ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧))))
11 df-po 5233 . 2 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
12 df-po 5233 . 2 (𝑆 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑆𝑥 ∧ ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
1310, 11, 123bitr4g 306 1 (𝑅 = 𝑆 → (𝑅 Po 𝐴𝑆 Po 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wral 3089   class class class wbr 4843   Po wpo 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-cleq 2792  df-clel 2795  df-ral 3094  df-br 4844  df-po 5233
This theorem is referenced by:  soeq1  5252
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