Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abpr Structured version   Visualization version   GIF version

Theorem abpr 42150
Description: Condition for a class abstraction to be a pair. (Contributed by RP, 25-Aug-2024.)
Assertion
Ref Expression
abpr ({𝑥𝜑} = {𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑌𝑥 = 𝑍)))
Distinct variable groups:   𝑥,𝑌   𝑥,𝑍
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abpr
StepHypRef Expression
1 dfpr2 4647 . 2 {𝑌, 𝑍} = {𝑥 ∣ (𝑥 = 𝑌𝑥 = 𝑍)}
21abeqabi 42149 1 ({𝑥𝜑} = {𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑌𝑥 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 845  wal 1539   = wceq 1541  {cab 2709  {cpr 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3953  df-sn 4629  df-pr 4631
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator