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Theorem distel 33779
Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5357 and elirrv 9355.) (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 5357 . . 3 𝑧 𝑥𝑧
2 df-ex 1783 . . . 4 (∃𝑧 𝑥𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥𝑧)
3 nfnae 2434 . . . . . 6 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 dveel1 2461 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥𝑧 → ∀𝑦 𝑥𝑧))
53, 4nf5d 2281 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥𝑧)
65nfnd 1861 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 ¬ 𝑥𝑧)
7 elequ2 2121 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
87notbid 318 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑥𝑧 ↔ ¬ 𝑥𝑦))
98a1i 11 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → (¬ 𝑥𝑧 ↔ ¬ 𝑥𝑦)))
103, 6, 9cbvald 2407 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑧 ¬ 𝑥𝑧 ↔ ∀𝑦 ¬ 𝑥𝑦))
1110notbid 318 . . . 4 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑧 ¬ 𝑥𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦))
122, 11syl5bb 283 . . 3 (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑧 𝑥𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦))
131, 12mpbii 232 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 ¬ 𝑥𝑦)
14 elirrv 9355 . . . . 5 ¬ 𝑦𝑦
15 elequ1 2113 . . . . 5 (𝑦 = 𝑥 → (𝑦𝑦𝑥𝑦))
1614, 15mtbii 326 . . . 4 (𝑦 = 𝑥 → ¬ 𝑥𝑦)
1716alimi 1814 . . 3 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 ¬ 𝑥𝑦)
1817con3i 154 . 2 (¬ ∀𝑦 ¬ 𝑥𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
1913, 18impbii 208 1 (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-reg 9351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564
This theorem is referenced by: (None)
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