Theorem raldifsnb 4766

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	⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)

Proof of Theorem raldifsnb

Step	Hyp	Ref	Expression
1		velsn 4607	. . . . . 6 ⊢ (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
2		nnel 3076	. . . . . 6 ⊢ (¬ 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌})
3		nne 2964	. . . . . 6 ⊢ (¬ 𝑥 ≠ 𝑌 ↔ 𝑥 = 𝑌)
4	1, 2, 3	3bitr4ri 307	. . . . 5 ⊢ (¬ 𝑥 ≠ 𝑌 ↔ ¬ 𝑥 ∉ {𝑌})
5	4	con4bii 324	. . . 4 ⊢ (𝑥 ≠ 𝑌 ↔ 𝑥 ∉ {𝑌})
6	5	imbi1i 352	. . 3 ⊢ ((𝑥 ≠ 𝑌 → 𝜑) ↔ (𝑥 ∉ {𝑌} → 𝜑))
7	6	ralbii 3113	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑))
8		raldifb 4103	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)
9	7, 8	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ∉ wnel 3066  ∀wral 3081   ∖ cdif 3903  {csn 4591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-nel 3067  df-ral 3082  df-v 3459  df-dif 3909  df-sn 4592

This theorem is used by:  dff14b  7274  isdomn5  20861  safesnsupfilb  44204