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Theorem dtruOLD 5445
Description: Obsolete version of dtru 5440 as of 1-Jan-2025. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2376. (Revised by BJ, 31-May-2019.) Avoid ax-12 2176. (Revised by Rohan Ridenour, 9-Oct-2024.) Use ax-pr 5431 instead of ax-pow 5364. (Revised by BTernaryTau, 3-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dtruOLD ¬ ∀𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruOLD
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el 5441 . . . 4 𝑤 𝑥𝑤
2 ax-nul 5305 . . . . 5 𝑧𝑥 ¬ 𝑥𝑧
3 elequ1 2114 . . . . . . 7 (𝑥 = 𝑤 → (𝑥𝑧𝑤𝑧))
43notbid 318 . . . . . 6 (𝑥 = 𝑤 → (¬ 𝑥𝑧 ↔ ¬ 𝑤𝑧))
54spw 2032 . . . . 5 (∀𝑥 ¬ 𝑥𝑧 → ¬ 𝑥𝑧)
62, 5eximii 1836 . . . 4 𝑧 ¬ 𝑥𝑧
7 exdistrv 1954 . . . 4 (∃𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧) ↔ (∃𝑤 𝑥𝑤 ∧ ∃𝑧 ¬ 𝑥𝑧))
81, 6, 7mpbir2an 711 . . 3 𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧)
9 ax9v2 2120 . . . . . 6 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
109com12 32 . . . . 5 (𝑥𝑤 → (𝑤 = 𝑧𝑥𝑧))
1110con3dimp 408 . . . 4 ((𝑥𝑤 ∧ ¬ 𝑥𝑧) → ¬ 𝑤 = 𝑧)
12112eximi 1835 . . 3 (∃𝑤𝑧(𝑥𝑤 ∧ ¬ 𝑥𝑧) → ∃𝑤𝑧 ¬ 𝑤 = 𝑧)
13 equequ2 2024 . . . . . . 7 (𝑧 = 𝑦 → (𝑤 = 𝑧𝑤 = 𝑦))
1413notbid 318 . . . . . 6 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 ↔ ¬ 𝑤 = 𝑦))
15 ax7v1 2008 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 = 𝑦𝑤 = 𝑦))
1615con3d 152 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
1716spimevw 1993 . . . . . 6 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
1814, 17biimtrdi 253 . . . . 5 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
19 ax7v1 2008 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
2019con3d 152 . . . . . . 7 (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
2120spimevw 1993 . . . . . 6 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
2221a1d 25 . . . . 5 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
2318, 22pm2.61i 182 . . . 4 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
2423exlimivv 1931 . . 3 (∃𝑤𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
258, 12, 24mp2b 10 . 2 𝑥 ¬ 𝑥 = 𝑦
26 exnal 1826 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2725, 26mpbi 230 1 ¬ ∀𝑥 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1537  wex 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779
This theorem is referenced by: (None)
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