Theorem absn 4587

Description: Condition for a class abstraction to be a singleton. Formerly part of proof of dfiota2 6455. (Contributed by Andrew Salmon, 30-Jun-2011.) (Revised by AV, 24-Aug-2022.)

Assertion

Ref	Expression
absn	⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))

Distinct variable group:   𝑥,𝑌

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem absn

Step	Hyp	Ref	Expression
1		df-sn 4568	. . 3 ⊢ {𝑌} = {𝑥 ∣ 𝑥 = 𝑌}
2	1	eqeq2i 2749	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌})
3		abbib 2805	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))
4	2, 3	bitri 275	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))

Syntax hints:   ↔ wb 206  ∀wal 1540   = wceq 1542  {cab 2714  {csn 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-sn 4568

This theorem is referenced by:  rabeqsn  4611  euabsn2  4669  dfiota2  6455  n0cut  28326  dfaiota2  47534  aiotaval  47543