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Theorem frege104 39100
 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 3430 . . 3 𝑍 ∈ V
3 eqeq1 2829 . . . 4 (𝑧 = 𝑍 → (𝑧 = 𝑋𝑍 = 𝑋))
4 eqeq2 2836 . . . 4 (𝑧 = 𝑍 → (𝑋 = 𝑧𝑋 = 𝑍))
53, 4imbi12d 336 . . 3 (𝑧 = 𝑍 → ((𝑧 = 𝑋𝑋 = 𝑧) ↔ (𝑍 = 𝑋𝑋 = 𝑍)))
6 frege55c 39051 . . 3 (𝑧 = 𝑋𝑋 = 𝑧)
72, 5, 6vtocl 3475 . 2 (𝑍 = 𝑋𝑋 = 𝑍)
81frege103 39099 . 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 1656   ∈ wcel 2164   ∪ cun 3796   class class class wbr 4875   I cid 5251  ‘cfv 6127  t+ctcl 14110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-frege1 38923  ax-frege2 38924  ax-frege8 38942  ax-frege52a 38990  ax-frege52c 39021 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ifp 1090  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353 This theorem is referenced by:  frege114  39110
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