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Theorem difrab2 31468
Description: Difference of two restricted class abstractions. Compare with difrab 4273. (Contributed by Thierry Arnoux, 3-Jan-2022.)
Assertion
Ref Expression
difrab2 ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Proof of Theorem difrab2
StepHypRef Expression
1 nfrab1 3429 . . 3 𝑥{𝑥𝐴𝜑}
2 nfrab1 3429 . . 3 𝑥{𝑥𝐵𝜑}
31, 2nfdif 4090 . 2 𝑥({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑})
4 nfrab1 3429 . 2 𝑥{𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
5 eldif 3925 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
65anbi1i 625 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑))
7 andi 1007 . . . . . . 7 ((𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ ((𝜑 ∧ ¬ 𝑥𝐵) ∨ (𝜑 ∧ ¬ 𝜑)))
8 pm3.24 404 . . . . . . . 8 ¬ (𝜑 ∧ ¬ 𝜑)
98biorfi 938 . . . . . . 7 ((𝜑 ∧ ¬ 𝑥𝐵) ↔ ((𝜑 ∧ ¬ 𝑥𝐵) ∨ (𝜑 ∧ ¬ 𝜑)))
10 ancom 462 . . . . . . 7 ((𝜑 ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐵𝜑))
117, 9, 103bitr2i 299 . . . . . 6 ((𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ (¬ 𝑥𝐵𝜑))
1211anbi2i 624 . . . . 5 ((𝑥𝐴 ∧ (𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑))) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵𝜑)))
13 anass 470 . . . . 5 (((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ (𝑥𝐴 ∧ (𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑))))
14 anass 470 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵𝜑)))
1512, 13, 143bitr4i 303 . . . 4 (((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑))
166, 15bitr4i 278 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
17 rabid 3430 . . 3 (𝑥 ∈ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
18 eldif 3925 . . . 4 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ (𝑥 ∈ {𝑥𝐴𝜑} ∧ ¬ 𝑥 ∈ {𝑥𝐵𝜑}))
19 rabid 3430 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
20 ianor 981 . . . . . 6 (¬ (𝑥𝐵𝜑) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝜑))
21 rabid 3430 . . . . . 6 (𝑥 ∈ {𝑥𝐵𝜑} ↔ (𝑥𝐵𝜑))
2220, 21xchnxbir 333 . . . . 5 𝑥 ∈ {𝑥𝐵𝜑} ↔ (¬ 𝑥𝐵 ∨ ¬ 𝜑))
2319, 22anbi12i 628 . . . 4 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ ¬ 𝑥 ∈ {𝑥𝐵𝜑}) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
2418, 23bitri 275 . . 3 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
2516, 17, 243bitr4ri 304 . 2 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑})
263, 4, 25eqri 3969 1 ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wo 846   = wceq 1542  wcel 2107  {crab 3410  cdif 3912
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3411  df-v 3450  df-dif 3918
This theorem is referenced by:  reprdifc  33280
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