Theorem abpr 44155

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

Assertion

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

Distinct variable groups:   𝑥,𝑌   𝑥,𝑍

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abpr

Step	Hyp	Ref	Expression
1		dfpr2 4610	. 2 ⊢ {𝑌, 𝑍} = {𝑥 ∣ (𝑥 = 𝑌 ∨ 𝑥 = 𝑍)}
2	1	abeqabi 44154	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑌 ∨ 𝑥 = 𝑍)))

Syntax hints:   ↔ wb 209   ∨ wo 860  ∀wal 1568   = wceq 1570  {cab 2741  {cpr 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-un 3910  df-sn 4590  df-pr 4592

This theorem is referenced by: (None)