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Theorem sn-dtru 39703
 Description: dtru 5240 without ax-8 2114 or ax-12 2176. (Contributed by SN, 21-Sep-2023.)
Assertion
Ref Expression
sn-dtru ¬ ∀𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem sn-dtru
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sn-el 39702 . . . 4 𝑤𝑥 𝑥𝑤
2 ax-nul 5177 . . . 4 𝑧𝑥 ¬ 𝑥𝑧
3 exdistrv 1957 . . . 4 (∃𝑤𝑧(∃𝑥 𝑥𝑤 ∧ ∀𝑥 ¬ 𝑥𝑧) ↔ (∃𝑤𝑥 𝑥𝑤 ∧ ∃𝑧𝑥 ¬ 𝑥𝑧))
41, 2, 3mpbir2an 711 . . 3 𝑤𝑧(∃𝑥 𝑥𝑤 ∧ ∀𝑥 ¬ 𝑥𝑧)
5 ax9v1 2124 . . . . . . . 8 (𝑤 = 𝑧 → (𝑥𝑤𝑥𝑧))
65eximdv 1919 . . . . . . 7 (𝑤 = 𝑧 → (∃𝑥 𝑥𝑤 → ∃𝑥 𝑥𝑧))
7 df-ex 1783 . . . . . . 7 (∃𝑥 𝑥𝑧 ↔ ¬ ∀𝑥 ¬ 𝑥𝑧)
86, 7syl6ib 254 . . . . . 6 (𝑤 = 𝑧 → (∃𝑥 𝑥𝑤 → ¬ ∀𝑥 ¬ 𝑥𝑧))
9 imnan 404 . . . . . 6 ((∃𝑥 𝑥𝑤 → ¬ ∀𝑥 ¬ 𝑥𝑧) ↔ ¬ (∃𝑥 𝑥𝑤 ∧ ∀𝑥 ¬ 𝑥𝑧))
108, 9sylib 221 . . . . 5 (𝑤 = 𝑧 → ¬ (∃𝑥 𝑥𝑤 ∧ ∀𝑥 ¬ 𝑥𝑧))
1110con2i 141 . . . 4 ((∃𝑥 𝑥𝑤 ∧ ∀𝑥 ¬ 𝑥𝑧) → ¬ 𝑤 = 𝑧)
12112eximi 1838 . . 3 (∃𝑤𝑧(∃𝑥 𝑥𝑤 ∧ ∀𝑥 ¬ 𝑥𝑧) → ∃𝑤𝑧 ¬ 𝑤 = 𝑧)
13 equeuclr 2031 . . . . . . 7 (𝑧 = 𝑦 → (𝑤 = 𝑦𝑤 = 𝑧))
1413con3d 155 . . . . . 6 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ¬ 𝑤 = 𝑦))
15 ax7v1 2018 . . . . . . . 8 (𝑥 = 𝑤 → (𝑥 = 𝑦𝑤 = 𝑦))
1615con3d 155 . . . . . . 7 (𝑥 = 𝑤 → (¬ 𝑤 = 𝑦 → ¬ 𝑥 = 𝑦))
1716spimevw 2002 . . . . . 6 𝑤 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
1814, 17syl6 35 . . . . 5 (𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
19 ax7v1 2018 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
2019con3d 155 . . . . . . 7 (𝑥 = 𝑧 → (¬ 𝑧 = 𝑦 → ¬ 𝑥 = 𝑦))
2120spimevw 2002 . . . . . 6 𝑧 = 𝑦 → ∃𝑥 ¬ 𝑥 = 𝑦)
2221a1d 25 . . . . 5 𝑧 = 𝑦 → (¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦))
2318, 22pm2.61i 185 . . . 4 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
2423exlimivv 1934 . . 3 (∃𝑤𝑧 ¬ 𝑤 = 𝑧 → ∃𝑥 ¬ 𝑥 = 𝑦)
254, 12, 24mp2b 10 . 2 𝑥 ¬ 𝑥 = 𝑦
26 exnal 1829 . 2 (∃𝑥 ¬ 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
2725, 26mpbi 233 1 ¬ ∀𝑥 𝑥 = 𝑦
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1537  ∃wex 1782 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2122  ax-nul 5177  ax-pow 5235 This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783 This theorem is referenced by: (None)
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