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Theorem freq1 5642
Description: Equality theorem for the well-founded predicate. (Contributed by NM, 9-Mar-1997.)
Assertion
Ref Expression
freq1 (𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))

Proof of Theorem freq1
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq 5144 . . . . . 6 (𝑅 = 𝑆 → (𝑧𝑅𝑦𝑧𝑆𝑦))
21notbid 318 . . . . 5 (𝑅 = 𝑆 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑆𝑦))
32rexralbidv 3215 . . . 4 (𝑅 = 𝑆 → (∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦 ↔ ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑆𝑦))
43imbi2d 340 . . 3 (𝑅 = 𝑆 → (((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑆𝑦)))
54albidv 1916 . 2 (𝑅 = 𝑆 → (∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦) ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑆𝑦)))
6 df-fr 5627 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑅𝑦))
7 df-fr 5627 . 2 (𝑆 Fr 𝐴 ↔ ∀𝑥((𝑥𝐴𝑥 ≠ ∅) → ∃𝑦𝑥𝑧𝑥 ¬ 𝑧𝑆𝑦))
85, 6, 73bitr4g 314 1 (𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wal 1532   = wceq 1534  wne 2935  wral 3056  wrex 3065  wss 3944  c0 4318   class class class wbr 5142   Fr wfr 5624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-br 5143  df-fr 5627
This theorem is referenced by:  weeq1  5660  freq12d  42385
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