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Theorem difrab2 29646
Description: Difference of two restricted class abstractions. Compare with difrab 4044. (Contributed by Thierry Arnoux, 3-Jan-2022.)
Assertion
Ref Expression
difrab2 ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}

Proof of Theorem difrab2
StepHypRef Expression
1 nfrab1 3261 . . 3 𝑥{𝑥𝐴𝜑}
2 nfrab1 3261 . . 3 𝑥{𝑥𝐵𝜑}
31, 2nfdif 3874 . 2 𝑥({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑})
4 nfrab1 3261 . 2 𝑥{𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
5 eldif 3725 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
65anbi1i 733 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑))
7 andi 947 . . . . . . 7 ((𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ ((𝜑 ∧ ¬ 𝑥𝐵) ∨ (𝜑 ∧ ¬ 𝜑)))
8 pm3.24 962 . . . . . . . 8 ¬ (𝜑 ∧ ¬ 𝜑)
98biorfi 421 . . . . . . 7 ((𝜑 ∧ ¬ 𝑥𝐵) ↔ ((𝜑 ∧ ¬ 𝑥𝐵) ∨ (𝜑 ∧ ¬ 𝜑)))
10 ancom 465 . . . . . . 7 ((𝜑 ∧ ¬ 𝑥𝐵) ↔ (¬ 𝑥𝐵𝜑))
117, 9, 103bitr2i 288 . . . . . 6 ((𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ (¬ 𝑥𝐵𝜑))
1211anbi2i 732 . . . . 5 ((𝑥𝐴 ∧ (𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑))) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵𝜑)))
13 anass 684 . . . . 5 (((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ (𝑥𝐴 ∧ (𝜑 ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑))))
14 anass 684 . . . . 5 (((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑) ↔ (𝑥𝐴 ∧ (¬ 𝑥𝐵𝜑)))
1512, 13, 143bitr4i 292 . . . 4 (((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)) ↔ ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ∧ 𝜑))
166, 15bitr4i 267 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝜑) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
17 rabid 3254 . . 3 (𝑥 ∈ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} ↔ (𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
18 eldif 3725 . . . 4 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ (𝑥 ∈ {𝑥𝐴𝜑} ∧ ¬ 𝑥 ∈ {𝑥𝐵𝜑}))
19 rabid 3254 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
20 rabid 3254 . . . . . . 7 (𝑥 ∈ {𝑥𝐵𝜑} ↔ (𝑥𝐵𝜑))
2120notbii 309 . . . . . 6 𝑥 ∈ {𝑥𝐵𝜑} ↔ ¬ (𝑥𝐵𝜑))
22 ianor 510 . . . . . 6 (¬ (𝑥𝐵𝜑) ↔ (¬ 𝑥𝐵 ∨ ¬ 𝜑))
2321, 22bitri 264 . . . . 5 𝑥 ∈ {𝑥𝐵𝜑} ↔ (¬ 𝑥𝐵 ∨ ¬ 𝜑))
2419, 23anbi12i 735 . . . 4 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ ¬ 𝑥 ∈ {𝑥𝐵𝜑}) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
2518, 24bitri 264 . . 3 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ ((𝑥𝐴𝜑) ∧ (¬ 𝑥𝐵 ∨ ¬ 𝜑)))
2616, 17, 253bitr4ri 293 . 2 (𝑥 ∈ ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) ↔ 𝑥 ∈ {𝑥 ∈ (𝐴𝐵) ∣ 𝜑})
273, 4, 26eqri 29624 1 ({𝑥𝐴𝜑} ∖ {𝑥𝐵𝜑}) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 382  wa 383   = wceq 1632  wcel 2139  {crab 3054  cdif 3712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718
This theorem is referenced by:  reprdifc  31014
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