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Theorem frege104 38578
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 3244 . . 3 𝑍 ∈ V
3 eqeq1 2655 . . . 4 (𝑧 = 𝑍 → (𝑧 = 𝑋𝑍 = 𝑋))
4 eqeq2 2662 . . . 4 (𝑧 = 𝑍 → (𝑋 = 𝑧𝑋 = 𝑍))
53, 4imbi12d 333 . . 3 (𝑧 = 𝑍 → ((𝑧 = 𝑋𝑋 = 𝑧) ↔ (𝑍 = 𝑋𝑋 = 𝑍)))
6 frege55c 38529 . . 3 (𝑧 = 𝑋𝑋 = 𝑧)
72, 5, 6vtocl 3290 . 2 (𝑍 = 𝑋𝑋 = 𝑍)
81frege103 38577 . 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 1523  wcel 2030  cun 3605   class class class wbr 4685   I cid 5052  cfv 5926  t+ctcl 13770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-frege1 38401  ax-frege2 38402  ax-frege8 38420  ax-frege52a 38468  ax-frege52c 38499
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150
This theorem is referenced by:  frege114  38588
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