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Theorem frege106 43430
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 43429 . 2 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
3 frege37 43301 . 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 1533  wcel 2098  cun 3947   class class class wbr 5152   I cid 5579  cfv 6553  t+ctcl 14972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-frege1 43251  ax-frege2 43252  ax-frege8 43270  ax-frege28 43291  ax-frege31 43295  ax-frege52a 43318
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689
This theorem is referenced by:  frege107  43431
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