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Theorem rabrab 3255
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 3254 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
21anbi1i 733 . . . 4 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓) ↔ ((𝑥𝐴𝜑) ∧ 𝜓))
3 anass 684 . . . 4 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
42, 3bitri 264 . . 3 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
54abbii 2877 . 2 {𝑥 ∣ (𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
6 df-rab 3059 . 2 {𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓)}
7 df-rab 3059 . 2 {𝑥𝐴 ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
85, 6, 73eqtr4i 2792 1 {𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wcel 2139  {cab 2746  {crab 3054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-rab 3059
This theorem is referenced by:  extwwlkfab  27532  clwwlknonclwlknonf1o  27543  dlwwlknondlwlknonf1o  27547  fpwrelmapffs  29839
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