Theorem raldifsnb 4768

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 4610	. . . . . 6 ⊢ (𝑥 ∈ {𝑌} ↔ 𝑥 = 𝑌)
2		nnel 3080	. . . . . 6 ⊢ (¬ 𝑥 ∉ {𝑌} ↔ 𝑥 ∈ {𝑌})
3		nne 2968	. . . . . 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 3117	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ≠ 𝑌 → 𝜑) ↔ ∀𝑥 ∈ 𝐴 (𝑥 ∉ {𝑌} → 𝜑))
8		raldifb 4111	. 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)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∉ wnel 3070  ∀wral 3085   ∖ cdif 3910  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-nel 3071  df-ral 3086  df-v 3465  df-dif 3916  df-sn 4595

This theorem is referenced by:  dff14b  7270  isdomn5  20794  safesnsupfilb  44035