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Theorem bj-inrab3 34150
Description: Generalization of dfrab3ss 4285, 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 4282 . . 3 {𝑥𝐵𝜑} = (𝐵 ∩ {𝑥𝜑})
21ineq2i 4190 . 2 (𝐴 ∩ {𝑥𝐵𝜑}) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
3 dfrab3 4282 . . . 4 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
43ineq2i 4190 . . 3 (𝐵 ∩ {𝑥𝐴𝜑}) = (𝐵 ∩ (𝐴 ∩ {𝑥𝜑}))
5 incom 4182 . . 3 ({𝑥𝐴𝜑} ∩ 𝐵) = (𝐵 ∩ {𝑥𝐴𝜑})
6 in12 4201 . . 3 (𝐴 ∩ (𝐵 ∩ {𝑥𝜑})) = (𝐵 ∩ (𝐴 ∩ {𝑥𝜑}))
74, 5, 63eqtr4i 2859 . 2 ({𝑥𝐴𝜑} ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ {𝑥𝜑}))
82, 7eqtr4i 2852 1 (𝐴 ∩ {𝑥𝐵𝜑}) = ({𝑥𝐴𝜑} ∩ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  {cab 2804  {crab 3147  cin 3939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-in 3947
This theorem is referenced by: (None)
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