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

Theorem clelab 2751
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 2622 . 2 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
2 nfv 1845 . . 3 𝑦(𝑥 = 𝐴𝜑)
3 nfv 1845 . . . 4 𝑥 𝑦 = 𝐴
4 nfsab1 2616 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfan 1830 . . 3 𝑥(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑})
6 eqeq1 2630 . . . 4 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
7 sbequ12 2113 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 df-clab 2613 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
97, 8syl6bbr 278 . . . 4 (𝑥 = 𝑦 → (𝜑𝑦 ∈ {𝑥𝜑}))
106, 9anbi12d 746 . . 3 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝜑) ↔ (𝑦 = 𝐴𝑦 ∈ {𝑥𝜑})))
112, 5, 10cbvex 2276 . 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 384   = wceq 1480  wex 1701  [wsb 1882  wcel 1992  {cab 2612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622
This theorem is referenced by:  elrabi  3347  bj-csbsnlem  32537  frege55c  37680  spr0nelg  41002
  Copyright terms: Public domain W3C validator