Theorem euabsn 3736

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 3735	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		nfv 1574	. . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥}
3		nfab1 2374	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
4	3	nfeq1 2382	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦}
5		sneq 3677	. . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
6	5	eqeq2d 2241	. . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	2, 4, 6	cbvex 1802	. 2 ⊢ (∃𝑥{𝑥 ∣ 𝜑} = {𝑥} ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
8	1, 7	bitr4i 187	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥{𝑥 ∣ 𝜑} = {𝑥})

Syntax hints:   ↔ wb 105   = wceq 1395  ∃wex 1538  ∃!weu 2077  {cab 2215  {csn 3666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sn 3672

This theorem is referenced by:  eusn  3740  args  5096  mapsn  6835