Theorem euabsn 3793

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)

Assertion

Ref	Expression
euabsn	⊢ (∃!xφ ↔ ∃x{x ∣ φ} = {x})

Proof of Theorem euabsn

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 3792	. 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y})
2		nfv 1619	. . 3 ⊢ Ⅎy{x ∣ φ} = {x}
3		nfab1 2492	. . . 4 ⊢ Ⅎx{x ∣ φ}
4	3	nfeq1 2499	. . 3 ⊢ Ⅎx{x ∣ φ} = {y}
5		sneq 3745	. . . 4 ⊢ (x = y → {x} = {y})
6	5	eqeq2d 2364	. . 3 ⊢ (x = y → ({x ∣ φ} = {x} ↔ {x ∣ φ} = {y}))
7	2, 4, 6	cbvex 1985	. 2 ⊢ (∃x{x ∣ φ} = {x} ↔ ∃y{x ∣ φ} = {y})
8	1, 7	bitr4i 243	1 ⊢ (∃!xφ ↔ ∃x{x ∣ φ} = {x})

Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642  ∃!weu 2204  {cab 2339  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sn 3742

This theorem is referenced by:  eusn  3797  uniintsn  3964  mapsn  6027