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Theorem abtp 42675
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 4686 . 2 {𝑋, 𝑌, 𝑍} = {𝑥 ∣ (𝑥 = 𝑋𝑥 = 𝑌𝑥 = 𝑍)}
21abeqabi 42673 1 ({𝑥𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋𝑥 = 𝑌𝑥 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  w3o 1083  wal 1531   = wceq 1533  {cab 2701  {ctp 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946  df-sn 4622  df-pr 4624  df-tp 4626
This theorem is referenced by: (None)
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