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Theorem bj-inrab2 36534
Description: Shorter proof of inrab 4305. (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 36533 . 2 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)}
2 nfv 1909 . . . 4 𝑥
3 inidm 4217 . . . . 5 (𝐴𝐴) = 𝐴
43a1i 11 . . . 4 (⊤ → (𝐴𝐴) = 𝐴)
52, 4rabeqd 3447 . . 3 (⊤ → {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)})
65mptru 1540 . 2 {𝑥 ∈ (𝐴𝐴) ∣ (𝜑𝜓)} = {𝑥𝐴 ∣ (𝜑𝜓)}
71, 6eqtri 2753 1 ({𝑥𝐴𝜑} ∩ {𝑥𝐴𝜓}) = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wtru 1534  {crab 3418  cin 3943
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-in 3951
This theorem is referenced by: (None)
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