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Theorem abtp 43365
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 4697 . 2 {𝑋, 𝑌, 𝑍} = {𝑥 ∣ (𝑥 = 𝑋𝑥 = 𝑌𝑥 = 𝑍)}
21abeqabi 43363 1 ({𝑥𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋𝑥 = 𝑌𝑥 = 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  w3o 1084  wal 1533   = wceq 1535  {cab 2710  {ctp 4635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1086  df-tru 1538  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-v 3479  df-un 3968  df-sn 4632  df-pr 4634  df-tp 4636
This theorem is referenced by: (None)
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