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Theorem raldifsnb 4795
Description: Restricted universal quantification on a class difference with a singleton in terms of an implication. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
Assertion
Ref Expression
raldifsnb (∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)

Proof of Theorem raldifsnb
StepHypRef Expression
1 velsn 4640 . . . . . 6 (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
2 nnel 3051 . . . . . 6 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌})
3 nne 2939 . . . . . 6 𝑥𝑌𝑥 = 𝑌)
41, 2, 33bitr4ri 304 . . . . 5 𝑥𝑌 ↔ ¬ 𝑥 ∉ {𝑌})
54con4bii 321 . . . 4 (𝑥𝑌𝑥 ∉ {𝑌})
65imbi1i 349 . . 3 ((𝑥𝑌𝜑) ↔ (𝑥 ∉ {𝑌} → 𝜑))
76ralbii 3088 . 2 (∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥𝐴 (𝑥 ∉ {𝑌} → 𝜑))
8 raldifb 4140 . 2 (∀𝑥𝐴 (𝑥 ∉ {𝑌} → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)
97, 8bitri 275 1 (∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1534  wcel 2099  wne 2935  wnel 3041  wral 3056  cdif 3941  {csn 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-nel 3042  df-ral 3057  df-v 3471  df-dif 3947  df-sn 4625
This theorem is referenced by:  dff14b  7275  isdomn5  21230  safesnsupfilb  42761
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