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

Theorem abbi 2720
Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
Assertion
Ref Expression
abbi (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})

Proof of Theorem abbi
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 hbab1 2595 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
2 hbab1 2595 . . 3 (𝑦 ∈ {𝑥𝜓} → ∀𝑥 𝑦 ∈ {𝑥𝜓})
31, 2cleqh 2707 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}))
4 abid 2594 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2594 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5bibi12i 327 . . 3 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1736 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
83, 7bitr2i 263 1 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wb 194  wal 1472   = wceq 1474  wcel 1976  {cab 2592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602
This theorem is referenced by:  abbii  2722  abbid  2723  nabbi  2880  rabbi  3093  sbcbi2  3447  rabeqsn  4157  iuneq12df  4471  dfiota2  5752  iotabi  5760  uniabio  5761  iotanul  5766  karden  8615  iuneq12daf  28559  bj-cleq  31942  abeq12  32934  elnev  37461  csbingVD  37942  csbsngVD  37951  csbxpgVD  37952  csbrngVD  37954  csbunigVD  37956  csbfv12gALTVD  37957
  Copyright terms: Public domain W3C validator