Theorem raldifb 4079

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 451	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∉ 𝐵 → 𝜑)))
2		df-nel 3050	. . . . . 6 ⊢ (𝑥 ∉ 𝐵 ↔ ¬ 𝑥 ∈ 𝐵)
3	2	anbi2i 623	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
4		eldif 3897	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 277	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
6	5	imbi1i 350	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) → 𝜑) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝜑))
7	1, 6	bitr3i 276	. 2 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∉ 𝐵 → 𝜑)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝜑))
8	7	ralbii2 3090	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   ∉ wnel 3049  ∀wral 3064   ∖ cdif 3884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nel 3050  df-ral 3069  df-v 3434  df-dif 3890

This theorem is referenced by:  raldifsnb  4729  coprmproddvdslem  16367  poimirlem26  35803  aacllem  46505