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Theorem frege106 37772
Description: Whatever follows 𝑋 in the 𝑅-sequence belongs to the 𝑅 -sequence beginning with 𝑋. Proposition 106 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege103.z 𝑍𝑉
Assertion
Ref Expression
frege106 (𝑋(t+‘𝑅)𝑍𝑋((t+‘𝑅) ∪ I )𝑍)

Proof of Theorem frege106
StepHypRef Expression
1 frege103.z . . 3 𝑍𝑉
21frege105 37771 . 2 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
3 frege37 37643 . 2 (((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍) → (𝑋(t+‘𝑅)𝑍𝑋((t+‘𝑅) ∪ I )𝑍))
42, 3ax-mp 5 1 (𝑋(t+‘𝑅)𝑍𝑋((t+‘𝑅) ∪ I )𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987  cun 3557   class class class wbr 4618   I cid 4989  cfv 5852  t+ctcl 13665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-frege1 37593  ax-frege2 37594  ax-frege8 37612  ax-frege28 37633  ax-frege31 37637  ax-frege52a 37660
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086
This theorem is referenced by:  frege107  37773
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