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Theorem freq2 5009
 Description: Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
freq2 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))

Proof of Theorem freq2
StepHypRef Expression
1 eqimss2 3621 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 frss 5005 . . 3 (𝐵𝐴 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
4 eqimss 3620 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 frss 5005 . . 3 (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
73, 6impbid 201 1 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ⊆ wss 3540   Fr wfr 4994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554  df-fr 4997 This theorem is referenced by:  weeq2  5027  frsn  5112  f1oweALT  7043  frfi  8090  freq12d  36627  ifr0  37675
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