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Theorem bj-inrab2 33418
 Description: Shorter proof of inrab 4099. (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 33417 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)}
2 nfv 2010 . . . 4 𝑥
3 inidm 4018 . . . . 5 (𝐴𝐴) = 𝐴
43a1i 11 . . . 4 (⊤ → (𝐴𝐴) = 𝐴)
52, 4bj-rabeqd 33409 . . 3 (⊤ → {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)})
65mptru 1661 . 2 {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)}
71, 6eqtri 2821 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 385   = wceq 1653  ⊤wtru 1654  {crab 3093   ∩ cin 3768 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-in 3776 This theorem is referenced by: (None)
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