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Theorem distel 35785
Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5448 and elirrv 9634.) (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 5448 . . 3 𝑧 𝑥𝑧
2 df-ex 1777 . . . 4 (∃𝑧 𝑥𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥𝑧)
3 nfnae 2437 . . . . . 6 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 dveel1 2464 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥𝑧 → ∀𝑦 𝑥𝑧))
53, 4nf5d 2283 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥𝑧)
65nfnd 1856 . . . . . 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 2410 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑧 ¬ 𝑥𝑧 ↔ ∀𝑦 ¬ 𝑥𝑦))
1110notbid 318 . . . 4 (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑧 ¬ 𝑥𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦))
122, 11bitrid 283 . . 3 (¬ ∀𝑦 𝑦 = 𝑥 → (∃𝑧 𝑥𝑧 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦))
131, 12mpbii 233 . 2 (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 ¬ 𝑥𝑦)
14 elirrv 9634 . . . . 5 ¬ 𝑦𝑦
15 elequ1 2113 . . . . 5 (𝑦 = 𝑥 → (𝑦𝑦𝑥𝑦))
1614, 15mtbii 326 . . . 4 (𝑦 = 𝑥 → ¬ 𝑥𝑦)
1716alimi 1808 . . 3 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 ¬ 𝑥𝑦)
1817con3i 154 . 2 (¬ ∀𝑦 ¬ 𝑥𝑦 → ¬ ∀𝑦 𝑦 = 𝑥)
1913, 18impbii 209 1 (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wal 1535  wex 1776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-reg 9630
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-sn 4632  df-pr 4634
This theorem is referenced by: (None)
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