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Theorem raldifsnb 4640
 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 4492 . . . . . 6 (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
2 nnel 3099 . . . . . 6 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌})
3 nne 2988 . . . . . 6 𝑥𝑌𝑥 = 𝑌)
41, 2, 33bitr4ri 305 . . . . 5 𝑥𝑌 ↔ ¬ 𝑥 ∉ {𝑌})
54con4bii 322 . . . 4 (𝑥𝑌𝑥 ∉ {𝑌})
65imbi1i 351 . . 3 ((𝑥𝑌𝜑) ↔ (𝑥 ∉ {𝑌} → 𝜑))
76ralbii 3132 . 2 (∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥𝐴 (𝑥 ∉ {𝑌} → 𝜑))
8 raldifb 4046 . 2 (∀𝑥𝐴 (𝑥 ∉ {𝑌} → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)
97, 8bitri 276 1 (∀𝑥𝐴 (𝑥𝑌𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   = wceq 1522   ∈ wcel 2081   ≠ wne 2984   ∉ wnel 3090  ∀wral 3105   ∖ cdif 3860  {csn 4476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-v 3439  df-dif 3866  df-sn 4477 This theorem is referenced by:  dff14b  6899
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