Theorem raldifb 4144

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 449	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∉ 𝐵 → 𝜑)))
2		df-nel 3037	. . . . . 6 ⊢ (𝑥 ∉ 𝐵 ↔ ¬ 𝑥 ∈ 𝐵)
3	2	anbi2i 621	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
4		eldif 3957	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 277	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
6	5	imbi1i 348	. . 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 3079	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∈ wcel 2099   ∉ wnel 3036  ∀wral 3051   ∖ cdif 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nel 3037  df-ral 3052  df-v 3464  df-dif 3950

This theorem is referenced by:  raldifsnb  4805  coprmproddvdslem  16663  poimirlem26  37347  aacllem  48549