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Theorem raldifsnb 4358
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)
Distinct variable group:   𝑥,𝑌
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem raldifsnb
StepHypRef Expression
1 velsn 4226 . . . . . 6 (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
2 nnel 2935 . . . . . 6 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌})
3 nne 2827 . . . . . 6 𝑥𝑌𝑥 = 𝑌)
41, 2, 33bitr4ri 293 . . . . 5 𝑥𝑌 ↔ ¬ 𝑥 ∉ {𝑌})
54con4bii 310 . . . 4 (𝑥𝑌𝑥 ∉ {𝑌})
65imbi1i 338 . . 3 ((𝑥𝑌𝜑) ↔ (𝑥 ∉ {𝑌} → 𝜑))
76ralbii 3009 . 2 (∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥𝐴 (𝑥 ∉ {𝑌} → 𝜑))
8 raldifb 3783 . 2 (∀𝑥𝐴 (𝑥 ∉ {𝑌} → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)
97, 8bitri 264 1 (∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1523  wcel 2030  wne 2823  wnel 2926  wral 2941  cdif 3604  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-v 3233  df-dif 3610  df-sn 4211
This theorem is referenced by:  dff14b  6568
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