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Theorem bj-inrab2 35102
Description: Shorter proof of inrab 4241. (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 35101 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)}
2 nfv 1917 . . . 4 𝑥
3 inidm 4153 . . . . 5 (𝐴𝐴) = 𝐴
43a1i 11 . . . 4 (⊤ → (𝐴𝐴) = 𝐴)
52, 4bj-rabeqd 35093 . . 3 (⊤ → {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)})
65mptru 1546 . 2 {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)}
71, 6eqtri 2766 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wtru 1540  {crab 3068  cin 3886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3432  df-in 3894
This theorem is referenced by: (None)
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