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Theorem bj-inrab2 32116
Description: Shorter proof of inrab 3858. (Contributed by BJ, 21-Apr-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inrab2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}

Proof of Theorem bj-inrab2
StepHypRef Expression
1 bj-inrab 32115 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)}
2 nfv 1830 . . . 4 𝑥
3 inidm 3784 . . . . 5 (𝐴𝐴) = 𝐴
43a1i 11 . . . 4 (⊤ → (𝐴𝐴) = 𝐴)
52, 4bj-rabeqd 32108 . . 3 (⊤ → {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)})
65trud 1484 . 2 {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)}
71, 6eqtri 2632 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1475  wtru 1476  {crab 2900  cin 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-in 3547
This theorem is referenced by: (None)
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