Theorem raldifsnb 4734

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 4582	. . . . . 6 ⊢ (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
2		nnel 3059	. . . . . 6 ⊢ (¬ 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌})
3		nne 2948	. . . . . 6 ⊢ (¬ 𝑥 ≠ 𝑌 ↔ 𝑥 = 𝑌)
4	1, 2, 3	3bitr4ri 303	. . . . 5 ⊢ (¬ 𝑥 ≠ 𝑌 ↔ ¬ 𝑥 ∉ {𝑌})
5	4	con4bii 320	. . . 4 ⊢ (𝑥 ≠ 𝑌 ↔ 𝑥 ∉ {𝑌})
6	5	imbi1i 349	. . 3 ⊢ ((𝑥 ≠ 𝑌 → 𝜑) ↔ (𝑥 ∉ {𝑌} → 𝜑))
7	6	ralbii 3092	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑))
8		raldifb 4083	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)
9	7, 8	bitri 274	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ {𝑌})𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2109   ≠ wne 2944   ∉ wnel 3050  ∀wral 3065   ∖ cdif 3888  {csn 4566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-nel 3051  df-ral 3070  df-v 3432  df-dif 3894  df-sn 4567

This theorem is referenced by:  dff14b  7138  isdomn5  40151