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Theorem weeq1 4121
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 4109 . . 3 (𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))
2 breq 3794 . . . . . . . 8 (𝑅 = 𝑆 → (𝑥𝑅𝑦𝑥𝑆𝑦))
3 breq 3794 . . . . . . . 8 (𝑅 = 𝑆 → (𝑦𝑅𝑧𝑦𝑆𝑧))
42, 3anbi12d 450 . . . . . . 7 (𝑅 = 𝑆 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑆𝑦𝑦𝑆𝑧)))
5 breq 3794 . . . . . . 7 (𝑅 = 𝑆 → (𝑥𝑅𝑧𝑥𝑆𝑧))
64, 5imbi12d 227 . . . . . 6 (𝑅 = 𝑆 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
76ralbidv 2343 . . . . 5 (𝑅 = 𝑆 → (∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
87ralbidv 2343 . . . 4 (𝑅 = 𝑆 → (∀𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦𝐴𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
98ralbidv 2343 . . 3 (𝑅 = 𝑆 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
101, 9anbi12d 450 . 2 (𝑅 = 𝑆 → ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (𝑆 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧))))
11 df-wetr 4099 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
12 df-wetr 4099 . 2 (𝑆 We 𝐴 ↔ (𝑆 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑆𝑦𝑦𝑆𝑧) → 𝑥𝑆𝑧)))
1310, 11, 123bitr4g 216 1 (𝑅 = 𝑆 → (𝑅 We 𝐴𝑆 We 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wral 2323   class class class wbr 3792   Fr wfr 4093   We wwe 4095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052  df-ral 2328  df-br 3793  df-frfor 4096  df-frind 4097  df-wetr 4099
This theorem is referenced by: (None)
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