Theorem raldifsni 4265

Description: Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Assertion

Ref	Expression
raldifsni	⊢ (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵))

Proof of Theorem raldifsni

Step	Hyp	Ref	Expression
1		eldifsn 4260	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	imbi1i 338	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑))
3		impexp 461	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑)))
4		df-ne 2782	. . . . . 6 ⊢ (𝑥 ≠ 𝐵 ↔ ¬ 𝑥 = 𝐵)
5	4	imbi1i 338	. . . . 5 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
6		con34b 305	. . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐵) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
7	5, 6	bitr4i 266	. . . 4 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (𝜑 → 𝑥 = 𝐵))
8	7	imbi2i 325	. . 3 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑)) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵)))
9	2, 3, 8	3bitri 285	. 2 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵)))
10	9	ralbii2 2961	1 ⊢ (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∖ cdif 3537  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-sn 4126

This theorem is referenced by:  islindf4  19996  snlindsntor  42054