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Theorem difrab2 30273
Description: Difference of two restricted class abstractions. Compare with difrab 4262. (Contributed by Thierry Arnoux, 3-Jan-2022.)
Assertion
Ref Expression
difrab2 ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Proof of Theorem difrab2
StepHypRef Expression
1 nfrab1 3375 . . 3 𝑥{𝑥𝐴𝜑}
2 nfrab1 3375 . . 3 𝑥{𝑥𝐵𝜑}
31, 2nfdif 4088 . 2 𝑥({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑})
4 nfrab1 3375 . 2 𝑥{𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
5 eldif 3929 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
65anbi1i 626 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑))
7 andi 1005 . . . . . . 7 ((𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ ((𝜑 ∧ ¬ 𝑥𝐵) ∨ (𝜑 ∧ ¬ 𝜑)))
8 pm3.24 406 . . . . . . . 8 ¬ (𝜑 ∧ ¬ 𝜑)
98biorfi 936 . . . . . . 7 ((𝜑 ∧ ¬ 𝑥𝐵) ↔ ((𝜑 ∧ ¬ 𝑥𝐵) ∨ (𝜑 ∧ ¬ 𝜑)))
10 ancom 464 . . . . . . 7 ((𝜑 ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐵𝜑))
117, 9, 103bitr2i 302 . . . . . 6 ((𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ (¬ 𝑥𝐵𝜑))
1211anbi2i 625 . . . . 5 ((𝑥𝐴 ∧ (𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑))) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵𝜑)))
13 anass 472 . . . . 5 (((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ (𝑥𝐴 ∧ (𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑))))
14 anass 472 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵𝜑)))
1512, 13, 143bitr4i 306 . . . 4 (((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑))
166, 15bitr4i 281 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
17 rabid 3369 . . 3 (𝑥 ∈ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
18 eldif 3929 . . . 4 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ (𝑥 ∈ {𝑥𝐴𝜑} ∧ ¬ 𝑥 ∈ {𝑥𝐵𝜑}))
19 rabid 3369 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
20 ianor 979 . . . . . 6 (¬ (𝑥𝐵𝜑) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝜑))
21 rabid 3369 . . . . . 6 (𝑥 ∈ {𝑥𝐵𝜑} ↔ (𝑥𝐵𝜑))
2220, 21xchnxbir 336 . . . . 5 𝑥 ∈ {𝑥𝐵𝜑} ↔ (¬ 𝑥𝐵 ∨ ¬ 𝜑))
2319, 22anbi12i 629 . . . 4 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ ¬ 𝑥 ∈ {𝑥𝐵𝜑}) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
2418, 23bitri 278 . . 3 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
2516, 17, 243bitr4ri 307 . 2 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑})
263, 4, 25eqri 3973 1 ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399  wo 844   = wceq 1538  wcel 2115  {crab 3137  cdif 3916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922
This theorem is referenced by:  reprdifc  31958
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