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Theorem weeq2 4094
 Description: Equality theorem for the well-ordering predicate. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
weeq2 (𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))

Proof of Theorem weeq2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 freq2 4083 . . 3 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 raleq 2505 . . . . 5 (𝐴 = 𝐵 → (∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
32raleqbi1dv 2513 . . . 4 (𝐴 = 𝐵 → (∀𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
43raleqbi1dv 2513 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
51, 4anbi12d 442 . 2 (𝐴 = 𝐵 → ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (𝑅 Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
6 df-wetr 4071 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
7 df-wetr 4071 . 2 (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
85, 6, 73bitr4g 212 1 (𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∀wral 2306   class class class wbr 3764   Fr wfr 4065   We wwe 4067 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-in 2924  df-ss 2931  df-frfor 4068  df-frind 4069  df-wetr 4071 This theorem is referenced by:  reg3exmid  4304
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