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Theorem freq2 5496
 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 3950 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 frss 5492 . . 3 (𝐵𝐴 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
4 eqimss 3949 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 frss 5492 . . 3 (𝐴𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Fr 𝐵𝑅 Fr 𝐴))
73, 6impbid 215 1 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1539   ⊆ wss 3859   Fr wfr 5481 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-in 3866  df-ss 3876  df-fr 5484 This theorem is referenced by:  weeq2  5514  frsn  5609  f1oweALT  7678  frfi  8789  freq12d  40349  ifr0  41520
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