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Theorem raldifsnb 4471
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 4337 . . . . . 6 (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
2 nnel 3044 . . . . . 6 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌})
3 nne 2936 . . . . . 6 𝑥𝑌𝑥 = 𝑌)
41, 2, 33bitr4ri 293 . . . . 5 𝑥𝑌 ↔ ¬ 𝑥 ∉ {𝑌})
54con4bii 310 . . . 4 (𝑥𝑌𝑥 ∉ {𝑌})
65imbi1i 338 . . 3 ((𝑥𝑌𝜑) ↔ (𝑥 ∉ {𝑌} → 𝜑))
76ralbii 3118 . 2 (∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥𝐴 (𝑥 ∉ {𝑌} → 𝜑))
8 raldifb 3893 . 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 1632  wcel 2139  wne 2932  wnel 3035  wral 3050  cdif 3712  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-v 3342  df-dif 3718  df-sn 4322
This theorem is referenced by:  dff14b  6692
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