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Theorem bj-ru0 34897
Description: The FOL part of Russell's paradox ru 3708 (see also bj-ru1 34898, bj-ru 34899). Use of elequ1 2118, bj-elequ12 34626 (instead of eleq1 2826, eleq12d 2833 as in ru 3708) permits to remove dependency on ax-10 2142, ax-11 2159, ax-12 2176, ax-ext 2709, df-sb 2072, df-clab 2716, df-cleq 2730, df-clel 2817. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ru0 ¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-ru0
StepHypRef Expression
1 pm5.19 391 . 2 ¬ (𝑦𝑦 ↔ ¬ 𝑦𝑦)
2 elequ1 2118 . . . 4 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
3 bj-elequ12 34626 . . . . . 6 ((𝑥 = 𝑦𝑥 = 𝑦) → (𝑥𝑥𝑦𝑦))
43anidms 570 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
54notbid 321 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
62, 5bibi12d 349 . . 3 (𝑥 = 𝑦 → ((𝑥𝑦 ↔ ¬ 𝑥𝑥) ↔ (𝑦𝑦 ↔ ¬ 𝑦𝑦)))
76spvv 2005 . 2 (∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥) → (𝑦𝑦 ↔ ¬ 𝑦𝑦))
81, 7mto 200 1 ¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wal 1541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788
This theorem is referenced by:  bj-ru1  34898
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