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Theorem raldifsni 4688
 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 4680 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
21imbi1i 353 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) → ¬ 𝜑))
3 impexp 454 . . 3 (((𝑥𝐴𝑥𝐵) → ¬ 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵 → ¬ 𝜑)))
4 df-ne 2952 . . . . . 6 (𝑥𝐵 ↔ ¬ 𝑥 = 𝐵)
54imbi1i 353 . . . . 5 ((𝑥𝐵 → ¬ 𝜑) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
6 con34b 319 . . . . 5 ((𝜑𝑥 = 𝐵) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
75, 6bitr4i 281 . . . 4 ((𝑥𝐵 → ¬ 𝜑) ↔ (𝜑𝑥 = 𝐵))
87imbi2i 339 . . 3 ((𝑥𝐴 → (𝑥𝐵 → ¬ 𝜑)) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝐵)))
92, 3, 83bitri 300 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝐵)))
109ralbii2 3095 1 (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070   ∖ cdif 3857  {csn 4525 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-v 3411  df-dif 3863  df-sn 4526 This theorem is referenced by:  islindf4  20617  snlindsntor  45304
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