Theorem raldifsni 4757

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 4747	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵))
2	1	imbi1i 352	. . 3 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑))
3		impexp 456	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑)))
4		df-ne 2959	. . . . . 6 ⊢ (𝑥 ≠ 𝐵 ↔ ¬ 𝑥 = 𝐵)
5	4	imbi1i 352	. . . . 5 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
6		con34b 319	. . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐵) ↔ (¬ 𝑥 = 𝐵 → ¬ 𝜑))
7	5, 6	bitr4i 281	. . . 4 ⊢ ((𝑥 ≠ 𝐵 → ¬ 𝜑) ↔ (𝜑 → 𝑥 = 𝐵))
8	7	imbi2i 339	. . 3 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ≠ 𝐵 → ¬ 𝜑)) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵)))
9	2, 3, 8	3bitri 300	. 2 ⊢ ((𝑥 ∈ (𝐴 ∖ {𝐵}) → ¬ 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 = 𝐵)))
10	9	ralbii2 3107	1 ⊢ (∀𝑥 ∈ (𝐴 ∖ {𝐵}) ¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  ∀wral 3079   ∖ cdif 4047  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-v 3455  df-dif 4049  df-sn 4588

This theorem is used by:  islindf4  22156  snlindsntor  49722