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Theorem clelab 2880
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-11 2155, see sbc5ALT 3804 for more details. (Revised by SN, 2-Sep-2024.)
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 elissetv 2815 . 2 (𝐴 ∈ {𝑥𝜑} → ∃𝑦 𝑦 = 𝐴)
2 exsimpl 1872 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑥 𝑥 = 𝐴)
3 eqeq1 2737 . . . 4 (𝑥 = 𝑦 → (𝑥 = 𝐴𝑦 = 𝐴))
43cbvexvw 2041 . . 3 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
52, 4sylib 217 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑦 𝑦 = 𝐴)
6 eleq1 2822 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
7 df-clab 2711 . . . . . 6 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
8 sb5 2268 . . . . . 6 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
97, 8bitri 275 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝑦𝜑))
10 eqeq2 2745 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
1110anbi1d 631 . . . . . 6 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
1211exbidv 1925 . . . . 5 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
139, 12bitrid 283 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
146, 13bitr3d 281 . . 3 (𝑦 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
1514exlimiv 1934 . 2 (∃𝑦 𝑦 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
161, 5, 15pm5.21nii 380 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  [wsb 2068  wcel 2107  {cab 2710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811
This theorem is referenced by:  elrabiOLD  3676  sbc5  3803  bj-csbsnlem  35688  frege55c  42540  spr0nelg  46017
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