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Theorem frege100 37277
 Description: One direction of dffrege99 37276. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege99.z 𝑍𝑈
Assertion
Ref Expression
frege100 (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))

Proof of Theorem frege100
StepHypRef Expression
1 frege99.z . . 3 𝑍𝑈
21dffrege99 37276 . 2 ((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
3 frege57aid 37186 . 2 (((¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋)))
42, 3ax-mp 5 1 (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑍 = 𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   class class class wbr 4583   I cid 4948  ‘cfv 5804  t+ctcl 13572 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-frege1 37104  ax-frege2 37105  ax-frege8 37123  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-id 4953  df-xp 5044  df-rel 5045 This theorem is referenced by:  frege101  37278  frege103  37280
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