Theorem raldifsni 4324

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 4317	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	imbi1i 339	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑))
3		impexp 462	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑)))
4		df-ne 2795	. . . . . 6 ⊢ (𝑥 ≠ 𝐵 ↔ ¬ 𝑥 = 𝐵)
5	4	imbi1i 339	. . . . 5 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
6		con34b 306	. . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐵) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
7	5, 6	bitr4i 267	. . . 4 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (𝜑 → 𝑥 = 𝐵))
8	7	imbi2i 326	. . 3 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑)) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵)))
9	2, 3, 8	3bitri 286	. 2 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵)))
10	9	ralbii2 2978	1 ⊢ (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912   ∖ cdif 3571  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-sn 4178

This theorem is referenced by:  islindf4  20177  snlindsntor  42260