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Theorem abbi 2724
 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 2599 . . 3 (𝑦 ∈ {𝑥𝜑} → ∀𝑥 𝑦 ∈ {𝑥𝜑})
2 hbab1 2599 . . 3 (𝑦 ∈ {𝑥𝜓} → ∀𝑥 𝑦 ∈ {𝑥𝜓})
31, 2cleqh 2711 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}))
4 abid 2598 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
5 abid 2598 . . . 4 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
64, 5bibi12i 328 . . 3 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}) ↔ (𝜑𝜓))
76albii 1737 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
83, 7bitr2i 264 1 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606 This theorem is referenced by:  abbii  2726  abbid  2727  nabbi  2884  rabbi  3097  sbcbi2  3451  rabeqsn  4161  iuneq12df  4480  dfiota2  5769  iotabi  5777  uniabio  5778  iotanul  5783  karden  8641  iuneq12daf  28756  bj-cleq  32142  abeq12  33134  elnev  37661  csbingVD  38142  csbsngVD  38151  csbxpgVD  38152  csbrngVD  38154  csbunigVD  38156  csbfv12gALTVD  38157
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