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Theorem weeq2 4340
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 4329 . . 3 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 raleq 2665 . . . . 5 (𝐴 = 𝐵 → (∀𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
32raleqbi1dv 2673 . . . 4 (𝐴 = 𝐵 → (∀𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
43raleqbi1dv 2673 . . 3 (𝐴 = 𝐵 → (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
51, 4anbi12d 470 . 2 (𝐴 = 𝐵 → ((𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (𝑅 Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
6 df-wetr 4317 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
7 df-wetr 4317 . 2 (𝑅 We 𝐵 ↔ (𝑅 Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
85, 6, 73bitr4g 222 1 (𝐴 = 𝐵 → (𝑅 We 𝐴𝑅 We 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wral 2448   class class class wbr 3987   Fr wfr 4311   We wwe 4313
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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-in 3127  df-ss 3134  df-frfor 4314  df-frind 4315  df-wetr 4317
This theorem is referenced by:  reg3exmid  4562
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