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Theorem rabrab 28510
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 2999 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
21anbi1i 726 . . . 4 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓) ↔ ((𝑥𝐴𝜑) ∧ 𝜓))
3 anass 678 . . . 4 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
42, 3bitri 262 . . 3 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
54abbii 2630 . 2 {𝑥 ∣ (𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
6 df-rab 2809 . 2 {𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓)}
7 df-rab 2809 . 2 {𝑥𝐴 ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
85, 6, 73eqtr4i 2546 1 {𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wcel 1938  {cab 2500  {crab 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-rab 2809
This theorem is referenced by:  fpwrelmapffs  28685
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