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Theorem frege75 37252
 Description: If from the proposition that 𝑥 has property 𝐴, whatever 𝑥 may be, it can be inferred that every result of an application of the procedure 𝑅 to 𝑥 has property 𝐴, then property 𝐴 is hereditary in the 𝑅-sequence. Proposition 75 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege75 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → 𝑅 hereditary 𝐴)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦

Proof of Theorem frege75
StepHypRef Expression
1 dffrege69 37246 . 2 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴)
2 frege52aid 37172 . 2 ((∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) ↔ 𝑅 hereditary 𝐴) → (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → 𝑅 hereditary 𝐴))
31, 2ax-mp 5 1 (∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)) → 𝑅 hereditary 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977   class class class wbr 4583   hereditary whe 37086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-frege52a 37171 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-he 37087 This theorem is referenced by:  frege97  37274  frege109  37286  frege131  37308
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