Theorem raldifsni 4744

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 4735	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	imbi1i 349	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑))
3		impexp 450	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑)))
4		df-ne 2929	. . . . . 6 ⊢ (𝑥 ≠ 𝐵 ↔ ¬ 𝑥 = 𝐵)
5	4	imbi1i 349	. . . . 5 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
6		con34b 316	. . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐵) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
7	5, 6	bitr4i 278	. . . 4 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (𝜑 → 𝑥 = 𝐵))
8	7	imbi2i 336	. . 3 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑)) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵)))
9	2, 3, 8	3bitri 297	. 2 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵)))
10	9	ralbii2 3074	1 ⊢ (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047   ∖ cdif 3894  {csn 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-v 3438  df-dif 3900  df-sn 4574

This theorem is referenced by:  islindf4  21775  snlindsntor  48582