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Theorem clelab 2207
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
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-clab 2070 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
21anbi2i 445 . . 3 ((𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}) ↔ (𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑))
32exbii 1537 . 2 (∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}) ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑))
4 df-clel 2079 . 2 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑦(𝑦 = 𝐴𝑦 ∈ {𝑥𝜑}))
5 nfv 1462 . . 3 𝑦(𝑥 = 𝐴𝜑)
6 nfv 1462 . . . 4 𝑥 𝑦 = 𝐴
7 nfs1v 1858 . . . 4 𝑥[𝑦 / 𝑥]𝜑
86, 7nfan 1498 . . 3 𝑥(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑)
9 eqeq1 2089 . . . 4 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
10 sbequ12 1696 . . . 4 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
119, 10anbi12d 457 . . 3 (𝑥 = 𝑦 → ((𝑥 = 𝐴𝜑) ↔ (𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑)))
125, 8, 11cbvex 1681 . 2 (∃𝑥(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑))
133, 4, 123bitr4i 210 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  [wsb 1687  {cab 2069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079
This theorem is referenced by:  elrabi  2755
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