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Theorem frege104 37743
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 3199 . . 3 𝑍 ∈ V
3 eqeq1 2625 . . . 4 (𝑧 = 𝑍 → (𝑧 = 𝑋𝑍 = 𝑋))
4 eqeq2 2632 . . . 4 (𝑧 = 𝑍 → (𝑋 = 𝑧𝑋 = 𝑍))
53, 4imbi12d 334 . . 3 (𝑧 = 𝑍 → ((𝑧 = 𝑋𝑋 = 𝑧) ↔ (𝑍 = 𝑋𝑋 = 𝑍)))
6 frege55c 37694 . . 3 (𝑧 = 𝑋𝑋 = 𝑧)
72, 5, 6vtocl 3245 . 2 (𝑍 = 𝑋𝑋 = 𝑍)
81frege103 37742 . 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 1480  wcel 1987  cun 3553   class class class wbr 4613   I cid 4984  cfv 5847  t+ctcl 13658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-frege1 37566  ax-frege2 37567  ax-frege8 37585  ax-frege52a 37633  ax-frege52c 37664
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081
This theorem is referenced by:  frege114  37753
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