Theorem abtp 43869

Description: Condition for a class abstraction to be a triple. (Contributed by RP, 25-Aug-2024.)

Assertion

Ref	Expression
abtp	⊢ ({𝑥 ∣ 𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍)))

Distinct variable groups:   𝑥,𝑋   𝑥,𝑌   𝑥,𝑍

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abtp

Step	Hyp	Ref	Expression
1		dftp2 4626	. 2 ⊢ {𝑋, 𝑌, 𝑍} = {𝑥 ∣ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍)}
2	1	abeqabi 43867	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍)))

Syntax hints:   ↔ wb 208   ∨ w3o 1092  ∀wal 1546   = wceq 1548  {cab 2719  {ctp 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-un 3890  df-sn 4559  df-pr 4561  df-tp 4563

This theorem is referenced by: (None)