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Theorem freq2 4111
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
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 frforeq2 4110 . . 3 (𝐴 = 𝐵 → ( FrFor 𝑅𝐴𝑠 ↔ FrFor 𝑅𝐵𝑠))
21albidv 1721 . 2 (𝐴 = 𝐵 → (∀𝑠 FrFor 𝑅𝐴𝑠 ↔ ∀𝑠 FrFor 𝑅𝐵𝑠))
3 df-frind 4097 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
4 df-frind 4097 . 2 (𝑅 Fr 𝐵 ↔ ∀𝑠 FrFor 𝑅𝐵𝑠)
52, 3, 43bitr4g 216 1 (𝐴 = 𝐵 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259   FrFor wfrfor 4092   Fr wfr 4093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-in 2952  df-ss 2959  df-frfor 4096  df-frind 4097
This theorem is referenced by:  weeq2  4122
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