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Theorem raldifsni 4462
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 4454 . . . 4 (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥𝐴𝑥𝐵))
21imbi1i 338 . . 3 ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ ((𝑥𝐴𝑥𝐵) → ¬ 𝜑))
3 impexp 437 . . 3 (((𝑥𝐴𝑥𝐵) → ¬ 𝜑) ↔ (𝑥𝐴 → (𝑥𝐵 → ¬ 𝜑)))
4 df-ne 2944 . . . . . 6 (𝑥𝐵 ↔ ¬ 𝑥 = 𝐵)
54imbi1i 338 . . . . 5 ((𝑥𝐵 → ¬ 𝜑) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
6 con34b 305 . . . . 5 ((𝜑𝑥 = 𝐵) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
75, 6bitr4i 267 . . . 4 ((𝑥𝐵 → ¬ 𝜑) ↔ (𝜑𝑥 = 𝐵))
87imbi2i 325 . . 3 ((𝑥𝐴 → (𝑥𝐵 → ¬ 𝜑)) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝐵)))
92, 3, 83bitri 286 . 2 ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ (𝑥𝐴 → (𝜑𝑥 = 𝐵)))
109ralbii2 3127 1 (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  cdif 3720  {csn 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-v 3353  df-dif 3726  df-sn 4318
This theorem is referenced by:  islindf4  20394  snlindsntor  42785
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