Mathbox for Wolf Lammen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-dfrabv Structured version   Visualization version   GIF version

Theorem wl-dfrabv 35026
 Description: Alternate definition of restricted class abstraction (df-wl-rab 35024), when 𝑥 and 𝐴 are disjoint. (Contributed by Wolf Lammen, 29-May-2023.)
Assertion
Ref Expression
wl-dfrabv {𝑥 : 𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem wl-dfrabv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wl-dfrabsb 35025 . 2 {𝑥 : 𝐴𝜑} = {𝑦 ∣ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)}
2 clelsb3 2920 . . . . . 6 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
3 clelsb3 2920 . . . . . 6 ([𝑧 / 𝑥]𝑥𝐴𝑧𝐴)
42, 3bitr4i 281 . . . . 5 ([𝑧 / 𝑦]𝑦𝐴 ↔ [𝑧 / 𝑥]𝑥𝐴)
5 sbco2vv 2106 . . . . 5 ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)
64, 5anbi12i 629 . . . 4 (([𝑧 / 𝑦]𝑦𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
7 df-clab 2780 . . . . 5 (𝑧 ∈ {𝑦 ∣ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ [𝑧 / 𝑦](𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑))
8 sban 2085 . . . . 5 ([𝑧 / 𝑦](𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑) ↔ ([𝑧 / 𝑦]𝑦𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
97, 8bitri 278 . . . 4 (𝑧 ∈ {𝑦 ∣ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ ([𝑧 / 𝑦]𝑦𝐴 ∧ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
10 df-clab 2780 . . . . 5 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ [𝑧 / 𝑥](𝑥𝐴𝜑))
11 sban 2085 . . . . 5 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
1210, 11bitri 278 . . . 4 (𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
136, 9, 123bitr4i 306 . . 3 (𝑧 ∈ {𝑦 ∣ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)} ↔ 𝑧 ∈ {𝑥 ∣ (𝑥𝐴𝜑)})
1413eqriv 2798 . 2 {𝑦 ∣ (𝑦𝐴 ∧ [𝑦 / 𝑥]𝜑)} = {𝑥 ∣ (𝑥𝐴𝜑)}
151, 14eqtri 2824 1 {𝑥 : 𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  [wsb 2069   ∈ wcel 2112  {cab 2779  {wl-crab 34999 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 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-wl-rab 35024 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator