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

Theorem abbi 2766
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 2640 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
2 hbab1 2640 . . 3 (𝑦 ∈ {𝑥𝜓} → ∀𝑥 𝑦 ∈ {𝑥𝜓})
31, 2cleqh 2753 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}))
4 abid 2639 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2639 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5bibi12i 328 . . 3 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1787 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
83, 7bitr2i 265 1 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1521   = wceq 1523  wcel 2030  {cab 2637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647
This theorem is referenced by:  abbii  2768  abbid  2769  nabbi  2925  rabbi  3150  sbcbi2  3517  rabeqsn  4246  iuneq12df  4576  dfiota2  5890  iotabi  5898  uniabio  5899  iotanul  5904  karden  8796  iuneq12daf  29499  bj-cleq  33074  abeq12  34094  elnev  38956  csbingVD  39434  csbsngVD  39443  csbxpgVD  39444  csbrngVD  39446  csbunigVD  39448  csbfv12gALTVD  39449
  Copyright terms: Public domain W3C validator