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Theorem rabdif 41502
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 4270 . 2 ({𝑥𝜑} ∩ (𝐴𝐵)) = (({𝑥𝜑} ∩ 𝐴) ∖ 𝐵)
2 dfrab2 4310 . 2 {𝑥 ∈ (𝐴𝐵) ∣ 𝜑} = ({𝑥𝜑} ∩ (𝐴𝐵))
3 dfrab2 4310 . . 3 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
43difeq1i 4118 . 2 ({𝑥𝐴𝜑} ∖ 𝐵) = (({𝑥𝜑} ∩ 𝐴) ∖ 𝐵)
51, 2, 43eqtr4ri 2770 1 ({𝑥𝐴𝜑} ∖ 𝐵) = {𝑥 ∈ (𝐴𝐵) ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  {cab 2708  {crab 3431  cdif 3945  cin 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-in 3955
This theorem is referenced by:  prjspeclsp  41820
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