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Theorem dffrege69 42278
Description: If from the proposition that 𝑥 has property 𝐴 it can be inferred generally, whatever 𝑥 may be, that every result of an application of the procedure 𝑅 to 𝑥 has property 𝐴, then we say " Property 𝐴 is hereditary in the 𝑅-sequence. Definition 69 of [Frege1879] p. 55. (Contributed by RP, 28-Mar-2020.)
Assertion
Ref Expression
dffrege69 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem dffrege69
StepHypRef Expression
1 dfhe3 42121 . 2 (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
21bicomi 223 1 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wcel 2107   class class class wbr 5110   hereditary whe 42118
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-he 42119
This theorem is referenced by:  frege70  42279  frege75  42284
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