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Theorem bj-inrab2 33049
Description: Shorter proof of inrab 3932. (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 33048 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)}
2 nfv 1883 . . . 4 𝑥
3 inidm 3855 . . . . 5 (𝐴𝐴) = 𝐴
43a1i 11 . . . 4 (⊤ → (𝐴𝐴) = 𝐴)
52, 4bj-rabeqd 33041 . . 3 (⊤ → {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)})
65trud 1533 . 2 {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)}
71, 6eqtri 2673 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wtru 1524  {crab 2945  cin 3606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-in 3614
This theorem is referenced by: (None)
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