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Theorem frege74 39072
Description: If 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then every result of a application of the procedure 𝑅 to 𝑋 has the property 𝐴. Proposition 74 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
frege74.x 𝑋𝑈
frege74.y 𝑌𝑉
Assertion
Ref Expression
frege74 (𝑋𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴)))

Proof of Theorem frege74
StepHypRef Expression
1 frege74.x . . 3 𝑋𝑈
2 frege74.y . . 3 𝑌𝑉
31, 2frege72 39070 . 2 (𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
4 ax-frege8 38944 . 2 ((𝑅 hereditary 𝐴 → (𝑋𝐴 → (𝑋𝑅𝑌𝑌𝐴))) → (𝑋𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴))))
53, 4ax-mp 5 1 (𝑋𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌𝑌𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166   class class class wbr 4874   hereditary whe 38907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128  ax-frege1 38925  ax-frege2 38926  ax-frege8 38944  ax-frege52a 38992  ax-frege58b 39036
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-xp 5349  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-he 38908
This theorem is referenced by:  frege81  39079
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