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

Theorem clelab 2887
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 2157, see sbc5ALT 3817 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 2822 . 2 (𝐴 ∈ {𝑥𝜑} → ∃𝑦 𝑦 = 𝐴)
2 exsimpl 1868 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑥 𝑥 = 𝐴)
3 iseqsetv-cleq 2806 . . 3 (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
42, 3sylib 218 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑦 𝑦 = 𝐴)
5 eleq1 2829 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
6 df-clab 2715 . . . . . 6 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
7 sb5 2276 . . . . . 6 ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
86, 7bitri 275 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝑦𝜑))
9 eqeq2 2749 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
109anbi1d 631 . . . . . 6 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
1110exbidv 1921 . . . . 5 (𝑦 = 𝐴 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
128, 11bitrid 283 . . . 4 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
135, 12bitr3d 281 . . 3 (𝑦 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
1413exlimiv 1930 . 2 (∃𝑦 𝑦 = 𝐴 → (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑)))
151, 4, 14pm5.21nii 378 1 (𝐴 ∈ {𝑥𝜑} ↔ ∃𝑥(𝑥 = 𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  [wsb 2064  wcel 2108  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816
This theorem is referenced by:  sbc5  3816  bj-csbsnlem  36904  frege55c  43931  spr0nelg  47463
  Copyright terms: Public domain W3C validator