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Theorem freq2 5417
 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 3947 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 frss 5413 . . 3 (𝐵𝐴 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
4 eqimss 3946 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 frss 5413 . . 3 (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
73, 6impbid 213 1 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   = wceq 1522   ⊆ wss 3861   Fr wfr 5402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-ext 2768 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-in 3868  df-ss 3876  df-fr 5405 This theorem is referenced by:  weeq2  5435  frsn  5528  f1oweALT  7532  frfi  8612  freq12d  39137  ifr0  40334
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