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Theorem rabdif 39113
Description: Move difference in and out of a restricted class abstraction. (Contributed by Steven Nguyen, 6-Jun-2023.)
Assertion
Ref Expression
rabdif ({𝑥𝐴𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabdif
StepHypRef Expression
1 indif2 4250 . 2 ({𝑥𝜑} ∩ (𝐴𝐵)) = (({𝑥𝜑} ∩ 𝐴) ∖ 𝐵)
2 dfrab2 4282 . 2 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = ({𝑥𝜑} ∩ (𝐴𝐵))
3 dfrab2 4282 . . 3 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
43difeq1i 4098 . 2 ({𝑥𝐴𝜑} ∖ 𝐵) = (({𝑥𝜑} ∩ 𝐴) ∖ 𝐵)
51, 2, 43eqtr4ri 2858 1 ({𝑥𝐴𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  {cab 2802  {crab 3145  cdif 3936  cin 3938
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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-in 3946
This theorem is referenced by:  prjspeclsp  39268
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