Theorem raldifb 4120

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 453	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) → 𝜑) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∉ 𝐵 → 𝜑)))
2		df-nel 3124	. . . . . 6 ⊢ (𝑥 ∉ 𝐵 ↔ ¬ 𝑥 ∈ 𝐵)
3	2	anbi2i 624	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
4		eldif 3945	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ∖ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5	3, 4	bitr4i 280	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) ↔ 𝑥 ∈ (𝐴 ∖ 𝐵))
6	5	imbi1i 352	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵) → 𝜑) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝜑))
7	1, 6	bitr3i 279	. 2 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∉ 𝐵 → 𝜑)) ↔ (𝑥 ∈ (𝐴 ∖ 𝐵) → 𝜑))
8	7	ralbii2 3163	1 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∉ 𝐵 → 𝜑) ↔ ∀𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110   ∉ wnel 3123  ∀wral 3138   ∖ cdif 3932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-nel 3124  df-ral 3143  df-v 3496  df-dif 3938

This theorem is referenced by:  raldifsnb  4728  coprmproddvdslem  16005  poimirlem26  34917  aacllem  44901