Theorem euabsn 3662

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 3661	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		nfv 1528	. . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥}
3		nfab1 2321	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
4	3	nfeq1 2329	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦}
5		sneq 3603	. . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
6	5	eqeq2d 2189	. . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	2, 4, 6	cbvex 1756	. 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 1353  ∃wex 1492  ∃!weu 2026  {cab 2163  {csn 3592

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sn 3598

This theorem is referenced by:  eusn  3666  args  4996  mapsn  6687