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Theorem bj-inrab3 34737
Description: Generalization of dfrab3ss 4199, which it may shorten. (Contributed by BJ, 21-Apr-2019.) (Revised by OpenAI, 7-Jul-2020.)
Assertion
Ref Expression
bj-inrab3 (𝐴 ∩ {𝑥𝐵𝜑}) = ({𝑥𝐴𝜑} ∩ 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bj-inrab3
StepHypRef Expression
1 dfrab3 4196 . . 3 {𝑥𝐵𝜑} = (𝐵 ∩ {𝑥𝜑})
21ineq2i 4098 . 2 (𝐴 ∩ {𝑥𝐵𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
3 dfrab3 4196 . . . 4 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
43ineq2i 4098 . . 3 (𝐵 ∩ {𝑥𝐴𝜑}) = (𝐵 ∩ (𝐴 ∩ {𝑥𝜑}))
5 incom 4089 . . 3 ({𝑥𝐴𝜑} ∩ 𝐵) = (𝐵 ∩ {𝑥𝐴𝜑})
6 in12 4109 . . 3 (𝐴 ∩ (𝐵 ∩ {𝑥𝜑})) = (𝐵 ∩ (𝐴 ∩ {𝑥𝜑}))
74, 5, 63eqtr4i 2771 . 2 ({𝑥𝐴𝜑} ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
82, 7eqtr4i 2764 1 (𝐴 ∩ {𝑥𝐵𝜑}) = ({𝑥𝐴𝜑} ∩ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  {cab 2716  {crab 3057  cin 3840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3399  df-in 3848
This theorem is referenced by: (None)
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