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Theorem raldifsni 4759
Description: Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Assertion
Ref Expression
raldifsni (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵))

Proof of Theorem raldifsni
StepHypRef Expression
1 eldifsn 4751 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
21imbi1i 350 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) → ¬ 𝜑))
3 impexp 452 . . 3 (((𝑥𝐴𝑥𝐵) → ¬ 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵 → ¬ 𝜑)))
4 df-ne 2941 . . . . . 6 (𝑥𝐵 ↔ ¬ 𝑥 = 𝐵)
54imbi1i 350 . . . . 5 ((𝑥𝐵 → ¬ 𝜑) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
6 con34b 316 . . . . 5 ((𝜑𝑥 = 𝐵) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
75, 6bitr4i 278 . . . 4 ((𝑥𝐵 → ¬ 𝜑) ↔ (𝜑𝑥 = 𝐵))
87imbi2i 336 . . 3 ((𝑥𝐴 → (𝑥𝐵 → ¬ 𝜑)) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝐵)))
92, 3, 83bitri 297 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝐵)))
109ralbii2 3089 1 (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  cdif 3911  {csn 4590
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-v 3449  df-dif 3917  df-sn 4591
This theorem is referenced by:  islindf4  21267  snlindsntor  46642
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