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Theorem weeq2 4388
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 4377 . . 3 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 raleq 2690 . . . . 5 (𝐴 = 𝐵 → (∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
32raleqbi1dv 2702 . . . 4 (𝐴 = 𝐵 → (∀𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
43raleqbi1dv 2702 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
51, 4anbi12d 473 . 2 (𝐴 = 𝐵 → ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (𝑅 Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
6 df-wetr 4365 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
7 df-wetr 4365 . 2 (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
85, 6, 73bitr4g 223 1 (𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wral 2472   class class class wbr 4029   Fr wfr 4359   We wwe 4361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-in 3159  df-ss 3166  df-frfor 4362  df-frind 4363  df-wetr 4365
This theorem is referenced by:  reg3exmid  4612
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