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

Theorem abtp 41756
Description: Condition for a class abstraction to be a triple. (Contributed by RP, 25-Aug-2024.)
Assertion
Ref Expression
abtp ({𝑥𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋𝑥 = 𝑌𝑥 = 𝑍)))
Distinct variable groups:   𝑥,𝑋   𝑥,𝑌   𝑥,𝑍
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abtp
StepHypRef Expression
1 dftp2 4655 . 2 {𝑋, 𝑌, 𝑍} = {𝑥 ∣ (𝑥 = 𝑋𝑥 = 𝑌𝑥 = 𝑍)}
21abeqabi 41754 1 ({𝑥𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋𝑥 = 𝑌𝑥 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  w3o 1087  wal 1540   = wceq 1542  {cab 2714  {ctp 4595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3450  df-un 3920  df-sn 4592  df-pr 4594  df-tp 4596
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator