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Theorem rabrab 28722
 Description: Abstract builder restricted to another restricted abstract builder. (Contributed by Thierry Arnoux, 30-Aug-2017.)
Assertion
Ref Expression
rabrab {𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜑𝜓)}

Proof of Theorem rabrab
StepHypRef Expression
1 rabid 3095 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
21anbi1i 727 . . . 4 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓) ↔ ((𝑥𝐴𝜑) ∧ 𝜓))
3 anass 679 . . . 4 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
42, 3bitri 263 . . 3 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
54abbii 2726 . 2 {𝑥 ∣ (𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
6 df-rab 2905 . 2 {𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓)}
7 df-rab 2905 . 2 {𝑥𝐴 ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
85, 6, 73eqtr4i 2642 1 {𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜑𝜓)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900 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-rab 2905 This theorem is referenced by:  fpwrelmapffs  28897
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