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Theorem difrab2 32005
Description: Difference of two restricted class abstractions. Compare with difrab 4307. (Contributed by Thierry Arnoux, 3-Jan-2022.)
Assertion
Ref Expression
difrab2 ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Proof of Theorem difrab2
StepHypRef Expression
1 nfrab1 3449 . . 3 𝑥{𝑥𝐴𝜑}
2 nfrab1 3449 . . 3 𝑥{𝑥𝐵𝜑}
31, 2nfdif 4124 . 2 𝑥({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑})
4 nfrab1 3449 . 2 𝑥{𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
5 eldif 3957 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
65anbi1i 622 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑))
7 andi 1004 . . . . . . 7 ((𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ ((𝜑 ∧ ¬ 𝑥𝐵) ∨ (𝜑 ∧ ¬ 𝜑)))
8 pm3.24 401 . . . . . . . 8 ¬ (𝜑 ∧ ¬ 𝜑)
98biorfi 935 . . . . . . 7 ((𝜑 ∧ ¬ 𝑥𝐵) ↔ ((𝜑 ∧ ¬ 𝑥𝐵) ∨ (𝜑 ∧ ¬ 𝜑)))
10 ancom 459 . . . . . . 7 ((𝜑 ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐵𝜑))
117, 9, 103bitr2i 298 . . . . . 6 ((𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ (¬ 𝑥𝐵𝜑))
1211anbi2i 621 . . . . 5 ((𝑥𝐴 ∧ (𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑))) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵𝜑)))
13 anass 467 . . . . 5 (((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ (𝑥𝐴 ∧ (𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑))))
14 anass 467 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵𝜑)))
1512, 13, 143bitr4i 302 . . . 4 (((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑))
166, 15bitr4i 277 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
17 rabid 3450 . . 3 (𝑥 ∈ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
18 eldif 3957 . . . 4 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ (𝑥 ∈ {𝑥𝐴𝜑} ∧ ¬ 𝑥 ∈ {𝑥𝐵𝜑}))
19 rabid 3450 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
20 ianor 978 . . . . . 6 (¬ (𝑥𝐵𝜑) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝜑))
21 rabid 3450 . . . . . 6 (𝑥 ∈ {𝑥𝐵𝜑} ↔ (𝑥𝐵𝜑))
2220, 21xchnxbir 332 . . . . 5 𝑥 ∈ {𝑥𝐵𝜑} ↔ (¬ 𝑥𝐵 ∨ ¬ 𝜑))
2319, 22anbi12i 625 . . . 4 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ ¬ 𝑥 ∈ {𝑥𝐵𝜑}) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
2418, 23bitri 274 . . 3 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
2516, 17, 243bitr4ri 303 . 2 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑})
263, 4, 25eqri 4001 1 ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 394  wo 843   = wceq 1539  wcel 2104  {crab 3430  cdif 3944
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rab 3431  df-v 3474  df-dif 3950
This theorem is referenced by:  reprdifc  33937
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