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Theorem freq2 5045
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 3637 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 frss 5041 . . 3 (𝐵𝐴 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
4 eqimss 3636 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 frss 5041 . . 3 (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
73, 6impbid 202 1 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wss 3555   Fr wfr 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-in 3562  df-ss 3569  df-fr 5033
This theorem is referenced by:  weeq2  5063  frsn  5150  f1oweALT  7097  frfi  8149  freq12d  37086  ifr0  38133
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