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Theorem weeq1 4278
Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 9-Mar-1997.)
Assertion
Ref Expression
weeq1 (𝑅 = 𝑆 → (𝑅 We 𝐴𝑆 We 𝐴))

Proof of Theorem weeq1
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 freq1 4266 . . 3 (𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))
2 breq 3931 . . . . . . . 8 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 breq 3931 . . . . . . . 8 (𝑅 = 𝑆 → (𝑦𝑅𝑧𝑦𝑆𝑧))
42, 3anbi12d 464 . . . . . . 7 (𝑅 = 𝑆 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑆𝑦𝑦𝑆𝑧)))
5 breq 3931 . . . . . . 7 (𝑅 = 𝑆 → (𝑥𝑅𝑧𝑥𝑆𝑧))
64, 5imbi12d 233 . . . . . 6 (𝑅 = 𝑆 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
76ralbidv 2437 . . . . 5 (𝑅 = 𝑆 → (∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
87ralbidv 2437 . . . 4 (𝑅 = 𝑆 → (∀𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦𝐴𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
98ralbidv 2437 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
101, 9anbi12d 464 . 2 (𝑅 = 𝑆 → ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (𝑆 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧))))
11 df-wetr 4256 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
12 df-wetr 4256 . 2 (𝑆 We 𝐴 ↔ (𝑆 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
1310, 11, 123bitr4g 222 1 (𝑅 = 𝑆 → (𝑅 We 𝐴𝑆 We 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wral 2416   class class class wbr 3929   Fr wfr 4250   We wwe 4252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-cleq 2132  df-clel 2135  df-ral 2421  df-br 3930  df-frfor 4253  df-frind 4254  df-wetr 4256
This theorem is referenced by: (None)
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