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

Theorem nabbi 3045
Description: Not equivalent wff's correspond to not equal class abstractions. (Contributed by AV, 7-Apr-2019.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Assertion
Ref Expression
nabbi (∃𝑥(𝜑 ↔ ¬ 𝜓) ↔ {𝑥𝜑} ≠ {𝑥𝜓})

Proof of Theorem nabbi
StepHypRef Expression
1 df-ne 2944 . 2 ({𝑥𝜑} ≠ {𝑥𝜓} ↔ ¬ {𝑥𝜑} = {𝑥𝜓})
2 exnal 1902 . . . 4 (∃𝑥 ¬ (𝜑𝜓) ↔ ¬ ∀𝑥(𝜑𝜓))
3 xor3 371 . . . . 5 (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))
43exbii 1924 . . . 4 (∃𝑥 ¬ (𝜑𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
52, 4bitr3i 266 . . 3 (¬ ∀𝑥(𝜑𝜓) ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
6 abbi 2886 . . 3 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
75, 6xchnxbi 321 . 2 (¬ {𝑥𝜑} = {𝑥𝜓} ↔ ∃𝑥(𝜑 ↔ ¬ 𝜓))
81, 7bitr2i 265 1 (∃𝑥(𝜑 ↔ ¬ 𝜓) ↔ {𝑥𝜑} ≠ {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1629   = wceq 1631  wex 1852  {cab 2757  wne 2943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ne 2944
This theorem is referenced by:  suppvalbr  7454
  Copyright terms: Public domain W3C validator