MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabrab Structured version   Visualization version   GIF version

Theorem rabrab 3360
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 3359 . . . . 5 (𝑥 ∈ {𝑥𝐴𝜑} ↔ (𝑥𝐴𝜑))
21anbi1i 626 . . . 4 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓) ↔ ((𝑥𝐴𝜑) ∧ 𝜓))
3 anass 472 . . . 4 (((𝑥𝐴𝜑) ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
42, 3bitri 278 . . 3 ((𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓) ↔ (𝑥𝐴 ∧ (𝜑𝜓)))
54abbii 2887 . 2 {𝑥 ∣ (𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
6 df-rab 3139 . 2 {𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥 ∣ (𝑥 ∈ {𝑥𝐴𝜑} ∧ 𝜓)}
7 df-rab 3139 . 2 {𝑥𝐴 ∣ (𝜑𝜓)} = {𝑥 ∣ (𝑥𝐴 ∧ (𝜑𝜓))}
85, 6, 73eqtr4i 2855 1 {𝑥 ∈ {𝑥𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2114  {cab 2800  {crab 3134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-rab 3139
This theorem is referenced by:  rabrabi  3468  extwwlkfab  28135  fpwrelmapffs  30480
  Copyright terms: Public domain W3C validator