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

Theorem rabrabi 3479
 Description: Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2146 and ax-11 2162. (Revised by Gino Giotto, 20-Aug-2023.)
Hypothesis
Ref Expression
rabrabi.1 (𝑥 = 𝑦 → (𝜒𝜑))
Assertion
Ref Expression
rabrabi {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥)

Proof of Theorem rabrabi
StepHypRef Expression
1 rabrabi.1 . . . 4 (𝑥 = 𝑦 → (𝜒𝜑))
21cbvrabv 3477 . . 3 {𝑥𝐴𝜒} = {𝑦𝐴𝜑}
32rabeqi 3467 . 2 {𝑥 ∈ {𝑥𝐴𝜒} ∣ 𝜓} = {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓}
4 rabrab 3370 . 2 {𝑥 ∈ {𝑥𝐴𝜒} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
53, 4eqtr3i 2849 1 {𝑥 ∈ {𝑦𝐴𝜑} ∣ 𝜓} = {𝑥𝐴 ∣ (𝜒𝜓)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  {crab 3137 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142 This theorem is referenced by:  wlksnwwlknvbij  27700
 Copyright terms: Public domain W3C validator