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Theorem raldifsnb 4796
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 4642 . . . . . 6 (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
2 nnel 3056 . . . . . 6 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌})
3 nne 2944 . . . . . 6 𝑥𝑌𝑥 = 𝑌)
41, 2, 33bitr4ri 304 . . . . 5 𝑥𝑌 ↔ ¬ 𝑥 ∉ {𝑌})
54con4bii 321 . . . 4 (𝑥𝑌𝑥 ∉ {𝑌})
65imbi1i 349 . . 3 ((𝑥𝑌𝜑) ↔ (𝑥 ∉ {𝑌} → 𝜑))
76ralbii 3093 . 2 (∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥𝐴 (𝑥 ∉ {𝑌} → 𝜑))
8 raldifb 4149 . 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 206   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wral 3061  cdif 3948  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-v 3482  df-dif 3954  df-sn 4627
This theorem is referenced by:  dff14b  7291  isdomn5  20710  safesnsupfilb  43431
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