MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abid2fOLD Structured version   Visualization version   GIF version

Theorem abid2fOLD 2939
Description: Obsolete version of abid2f 2938 as of 26-Feb-2025. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
abid2f.1 𝑥𝐴
Assertion
Ref Expression
abid2fOLD {𝑥𝑥𝐴} = 𝐴

Proof of Theorem abid2fOLD
StepHypRef Expression
1 nfab1 2906 . . 3 𝑥{𝑥𝑥𝐴}
2 abid2f.1 . . 3 𝑥𝐴
31, 2cleqf 2936 . 2 ({𝑥𝑥𝐴} = 𝐴 ↔ ∀𝑥(𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴))
4 abid 2715 . 2 (𝑥 ∈ {𝑥𝑥𝐴} ↔ 𝑥𝐴)
53, 4mpgbir 1797 1 {𝑥𝑥𝐴} = 𝐴
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wcel 2103  {cab 2711  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator