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Theorem clelab 2928
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 2798 . 2 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
2 nfv 2005 . . 3 𝑦(𝑥 = 𝐴𝜑)
3 nfv 2005 . . . 4 𝑥 𝑦 = 𝐴
4 nfsab1 2792 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfan 1990 . . 3 𝑥(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑})
6 eqeq1 2806 . . . 4 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
7 sbequ12 2278 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 df-clab 2789 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
97, 8syl6bbr 280 . . . 4 (𝑥 = 𝑦 → (𝜑𝑦 ∈ {𝑥𝜑}))
106, 9anbi12d 618 . . 3 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝜑) ↔ (𝑦 = 𝐴𝑦 ∈ {𝑥𝜑})))
112, 5, 10cbvex 2444 . 2 (∃𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
121, 11bitr4i 269 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wex 1859  [wsb 2059  wcel 2155  {cab 2788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798
This theorem is referenced by:  elrabi  3550  bj-csbsnlem  33200  frege55c  38706  spr0nelg  42288
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