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Theorem frege105 43932
Description: Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege103.z 𝑍𝑉
Assertion
Ref Expression
frege105 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)

Proof of Theorem frege105
StepHypRef Expression
1 frege103.z . . 3 𝑍𝑉
21dffrege99 43926 . 2 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
3 frege52aid 43822 . 2 (((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍) → ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍))
42, 3ax-mp 5 1 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  wcel 2108  cun 3974   class class class wbr 5166   I cid 5592  cfv 6575  t+ctcl 15036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-frege52a 43821
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707
This theorem is referenced by:  frege106  43933  frege112  43939
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