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Theorem frege104 42229
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 3464 . . 3 𝑍 ∈ V
3 eqeq1 2740 . . . 4 (𝑧 = 𝑍 → (𝑧 = 𝑋𝑍 = 𝑋))
4 eqeq2 2748 . . . 4 (𝑧 = 𝑍 → (𝑋 = 𝑧𝑋 = 𝑍))
53, 4imbi12d 344 . . 3 (𝑧 = 𝑍 → ((𝑧 = 𝑋𝑋 = 𝑧) ↔ (𝑍 = 𝑋𝑋 = 𝑍)))
6 frege55c 42180 . . 3 (𝑧 = 𝑋𝑋 = 𝑧)
72, 5, 6vtocl 3518 . 2 (𝑍 = 𝑋𝑋 = 𝑍)
81frege103 42228 . 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 1541  wcel 2106  cun 3908   class class class wbr 5105   I cid 5530  cfv 6496  t+ctcl 14870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-frege1 42052  ax-frege2 42053  ax-frege8 42071  ax-frege52a 42119  ax-frege52c 42150
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640
This theorem is referenced by:  frege114  42239
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