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Theorem distel 31407
Description: Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4807 and elirrv 8448.) (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 4807 . . 3 𝑧 𝑥𝑧
2 df-ex 1702 . . . 4 (∃𝑧 𝑥𝑧 ↔ ¬ ∀𝑧 ¬ 𝑥𝑧)
3 nfnae 2317 . . . . . 6 𝑦 ¬ ∀𝑦 𝑦 = 𝑥
4 dveel1 2369 . . . . . . . 8 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥𝑧 → ∀𝑦 𝑥𝑧))
53, 4nf5d 2115 . . . . . . 7 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥𝑧)
65nfnd 1782 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 ¬ 𝑥𝑧)
7 elequ2 2001 . . . . . . . 8 (𝑧 = 𝑦 → (𝑥𝑧𝑥𝑦))
87notbid 308 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑥𝑧 ↔ ¬ 𝑥𝑦))
98a1i 11 . . . . . 6 (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → (¬ 𝑥𝑧 ↔ ¬ 𝑥𝑦)))
103, 6, 9cbvald 2276 . . . . 5 (¬ ∀𝑦 𝑦 = 𝑥 → (∀𝑧 ¬ 𝑥𝑧 ↔ ∀𝑦 ¬ 𝑥𝑦))
1110notbid 308 . . . 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 8448 . . . . 5 ¬ 𝑦𝑦
15 elequ1 1994 . . . . 5 (𝑦 = 𝑥 → (𝑦𝑦𝑥𝑦))
1614, 15mtbii 316 . . . 4 (𝑦 = 𝑥 → ¬ 𝑥𝑦)
1716alimi 1736 . . 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 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-reg 8441
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-pr 4151
This theorem is referenced by: (None)
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