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

Theorem clelab 2818
 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 df-clel 2688 . 2 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
2 nfv 1924 . . 3 𝑦(𝑥 = 𝐴𝜑)
3 nfv 1924 . . . 4 𝑥 𝑦 = 𝐴
4 nfsab1 2682 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfan 1909 . . 3 𝑥(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑})
6 eqeq1 2696 . . . 4 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
7 sbequ12 2190 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 df-clab 2679 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
97, 8syl6bbr 278 . . . 4 (𝑥 = 𝑦 → (𝜑𝑦 ∈ {𝑥𝜑}))
106, 9anbi12d 749 . . 3 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝜑) ↔ (𝑦 = 𝐴𝑦 ∈ {𝑥𝜑})))
112, 5, 10cbvex 2349 . 2 (∃𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
121, 11bitr4i 267 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1564  ∃wex 1785  [wsb 1978   ∈ wcel 2071  {cab 2678 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-clab 2679  df-cleq 2685  df-clel 2688 This theorem is referenced by:  elrabi  3432  bj-csbsnlem  33093  frege55c  38599  spr0nelg  42121
 Copyright terms: Public domain W3C validator