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Theorem distel 31833
Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4877 and elirrv 8542.) (Contributed by Scott Fenton, 15-Dec-2010.)
Assertion
Ref Expression
distel (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦)

Proof of Theorem distel
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 el 4877 . . 3 𝑧 𝑥𝑧
2 df-ex 1745 . . . 4 (∃𝑧 𝑥𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥𝑧)
3 nfnae 2351 . . . . . 6 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 dveel1 2398 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥𝑧 → ∀𝑦 𝑥𝑧))
53, 4nf5d 2156 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥𝑧)
65nfnd 1825 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 ¬ 𝑥𝑧)
7 elequ2 2044 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
87notbid 307 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑥𝑧 ↔ ¬ 𝑥𝑦))
98a1i 11 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → (¬ 𝑥𝑧 ↔ ¬ 𝑥𝑦)))
103, 6, 9cbvald 2313 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑧 ¬ 𝑥𝑧 ↔ ∀𝑦 ¬ 𝑥𝑦))
1110notbid 307 . . . 4 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑧 ¬ 𝑥𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦))
122, 11syl5bb 272 . . 3 (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑧 𝑥𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦))
131, 12mpbii 223 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 ¬ 𝑥𝑦)
14 elirrv 8542 . . . . 5 ¬ 𝑦𝑦
15 elequ1 2037 . . . . 5 (𝑦 = 𝑥 → (𝑦𝑦𝑥𝑦))
1614, 15mtbii 315 . . . 4 (𝑦 = 𝑥 → ¬ 𝑥𝑦)
1716alimi 1779 . . 3 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 ¬ 𝑥𝑦)
1817con3i 150 . 2 (¬ ∀𝑦 ¬ 𝑥𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
1913, 18impbii 199 1 (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by: (None)
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