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Theorem raldifsni 4798
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 4790 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
21imbi1i 349 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) → ¬ 𝜑))
3 impexp 451 . . 3 (((𝑥𝐴𝑥𝐵) → ¬ 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵 → ¬ 𝜑)))
4 df-ne 2941 . . . . . 6 (𝑥𝐵 ↔ ¬ 𝑥 = 𝐵)
54imbi1i 349 . . . . 5 ((𝑥𝐵 → ¬ 𝜑) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
6 con34b 315 . . . . 5 ((𝜑𝑥 = 𝐵) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
75, 6bitr4i 277 . . . 4 ((𝑥𝐵 → ¬ 𝜑) ↔ (𝜑𝑥 = 𝐵))
87imbi2i 335 . . 3 ((𝑥𝐴 → (𝑥𝐵 → ¬ 𝜑)) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝐵)))
92, 3, 83bitri 296 . 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 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  cdif 3945  {csn 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-v 3476  df-dif 3951  df-sn 4629
This theorem is referenced by:  islindf4  21392  snlindsntor  47142
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