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Theorem clelab 2886
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 2756 . 2 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
2 nfv 1992 . . 3 𝑦(𝑥 = 𝐴𝜑)
3 nfv 1992 . . . 4 𝑥 𝑦 = 𝐴
4 nfsab1 2750 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
53, 4nfan 1977 . . 3 𝑥(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑})
6 eqeq1 2764 . . . 4 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
7 sbequ12 2258 . . . . 5 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8 df-clab 2747 . . . . 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 2417 . 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 1632  wex 1853  [wsb 2046  wcel 2139  {cab 2746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756
This theorem is referenced by:  elrabi  3499  bj-csbsnlem  33204  frege55c  38714  spr0nelg  42236
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