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

Theorem clelab 2914
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
Assertion
Ref Expression
clelab (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem clelab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfclel 2848 . 2 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
2 nfv 1873 . . 3 𝑦(𝑥 = 𝐴𝜑)
3 nfv 1873 . . . 4 𝑥 𝑦 = 𝐴
4 nfsab1 2768 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfan 1862 . . 3 𝑥(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑})
6 eqeq1 2783 . . . 4 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
7 sbequ12 2179 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 df-clab 2760 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
97, 8syl6bbr 281 . . . 4 (𝑥 = 𝑦 → (𝜑𝑦 ∈ {𝑥𝜑}))
106, 9anbi12d 621 . . 3 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝜑) ↔ (𝑦 = 𝐴𝑦 ∈ {𝑥𝜑})))
112, 5, 10cbvexv1 2278 . 2 (∃𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
121, 11bitr4i 270 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  wex 1742  [wsb 2015  wcel 2050  {cab 2759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847
This theorem is referenced by:  elrabi  3591  bj-csbsnlem  33709  frege55c  39624  spr0nelg  43004
  Copyright terms: Public domain W3C validator