Theorem raldifb 4124

Description: Restricted universal quantification on a class difference in terms of an implication. (Contributed by Alexander van der Vekens, 3-Jan-2018.)

Assertion

Ref	Expression
raldifb	⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)

Proof of Theorem raldifb

Step	Hyp	Ref	Expression
1		impexp 450	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∉ 𝐵 → 𝜑)))
2		df-nel 3037	. . . . . 6 ⊢ (𝑥 ∉ 𝐵 ↔ ¬ 𝑥 ∈ 𝐵)
3	2	anbi2i 623	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
4		eldif 3936	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 278	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
6	5	imbi1i 349	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) → 𝜑) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝜑))
7	1, 6	bitr3i 277	. 2 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∉ 𝐵 → 𝜑)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝜑))
8	7	ralbii2 3078	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ∉ wnel 3036  ∀wral 3051   ∖ cdif 3923

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nel 3037  df-ral 3052  df-v 3461  df-dif 3929

This theorem is referenced by:  raldifsnb  4772  coprmproddvdslem  16681  poimirlem26  37670  aacllem  49665