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Theorem abtp 43423
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 4691 . 2 {𝑋, 𝑌, 𝑍} = {𝑥 ∣ (𝑥 = 𝑋𝑥 = 𝑌𝑥 = 𝑍)}
21abeqabi 43421 1 ({𝑥𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋𝑥 = 𝑌𝑥 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  w3o 1086  wal 1538   = wceq 1540  {cab 2714  {ctp 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-pr 4629  df-tp 4631
This theorem is referenced by: (None)
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