Theorem euabsn 3777

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 3776	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		nfv 1581	. . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥}
3		nfab1 2394	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
4	3	nfeq1 2402	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦}
5		sneq 3716	. . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
6	5	eqeq2d 2250	. . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	2, 4, 6	cbvex 1809	. 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 1402  ∃wex 1545  ∃!weu 2086  {cab 2224  {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3711

This theorem is referenced by:  eusn  3781  args  5151  mapsnd  6960  mapsn  6962