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Theorem freq2 5519
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 4021 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 frss 5515 . . 3 (𝐵𝐴 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
4 eqimss 4020 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 frss 5515 . . 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 1528  wss 3933   Fr wfr 5504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-in 3940  df-ss 3949  df-fr 5507
This theorem is referenced by:  weeq2  5537  frsn  5632  f1oweALT  7662  frfi  8751  freq12d  39517  ifr0  40659
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