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Theorem frege104 43931
Description: Proposition 104 of [Frege1879] p. 73.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)

Hypothesis
Ref Expression
frege103.z 𝑍𝑉
Assertion
Ref Expression
frege104 (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍))

Proof of Theorem frege104
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 frege103.z . . . 4 𝑍𝑉
21elexi 3511 . . 3 𝑍 ∈ V
3 eqeq1 2744 . . . 4 (𝑧 = 𝑍 → (𝑧 = 𝑋𝑍 = 𝑋))
4 eqeq2 2752 . . . 4 (𝑧 = 𝑍 → (𝑋 = 𝑧𝑋 = 𝑍))
53, 4imbi12d 344 . . 3 (𝑧 = 𝑍 → ((𝑧 = 𝑋𝑋 = 𝑧) ↔ (𝑍 = 𝑋𝑋 = 𝑍)))
6 frege55c 43882 . . 3 (𝑧 = 𝑋𝑋 = 𝑧)
72, 5, 6vtocl 3570 . 2 (𝑍 = 𝑋𝑋 = 𝑍)
81frege103 43930 . 2 ((𝑍 = 𝑋𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍)))
97, 8ax-mp 5 1 (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍𝑋 = 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = 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-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-frege1 43754  ax-frege2 43755  ax-frege8 43773  ax-frege52a 43821  ax-frege52c 43852
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-nf 1782  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-sbc 3805  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:  frege114  43941
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