Theorem euabsn 4659

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 4658	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		nfv 1918	. . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥}
3		nfab1 2908	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
4	3	nfeq1 2921	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦}
5		sneq 4568	. . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
6	5	eqeq2d 2749	. . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	2, 4, 6	cbvexv1 2341	. 2 ⊢ (∃𝑥{𝑥 ∣ 𝜑} = {𝑥} ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
8	1, 7	bitr4i 277	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥{𝑥 ∣ 𝜑} = {𝑥})

Syntax hints:   ↔ wb 205   = wceq 1539  ∃wex 1783  ∃!weu 2568  {cab 2715  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-nfc 2888  df-sn 4559

This theorem is referenced by:  eusn  4663  uniintsn  4915  args  5989  opabiotadm  6832  mapsnd  8632