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Theorem abbi 2734
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 2610 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
2 hbab1 2610 . . 3 (𝑦 ∈ {𝑥𝜓} → ∀𝑥 𝑦 ∈ {𝑥𝜓})
31, 2cleqh 2721 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}))
4 abid 2609 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2609 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5bibi12i 329 . . 3 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1744 . 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 1478   = wceq 1480  wcel 1987  {cab 2607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617
This theorem is referenced by:  abbii  2736  abbid  2737  nabbi  2892  rabbi  3113  sbcbi2  3471  rabeqsn  4192  iuneq12df  4517  dfiota2  5821  iotabi  5829  uniabio  5830  iotanul  5835  karden  8718  iuneq12daf  29260  bj-cleq  32649  abeq12  33635  elnev  38160  csbingVD  38642  csbsngVD  38651  csbxpgVD  38652  csbrngVD  38654  csbunigVD  38656  csbfv12gALTVD  38657
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