Theorem euabsn 4654

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 4653	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		nfv 1909	. . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥}
3		nfab1 2977	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
4	3	nfeq1 2991	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦}
5		sneq 4569	. . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
6	5	eqeq2d 2830	. . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦}))
7	2, 4, 6	cbvexv1 2356	. 2 ⊢ (∃𝑥{𝑥 ∣ 𝜑} = {𝑥} ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
8	1, 7	bitr4i 280	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥{𝑥 ∣ 𝜑} = {𝑥})

Syntax hints:   ↔ wb 208   = wceq 1531  ∃wex 1774  ∃!weu 2647  {cab 2797  {csn 4559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-nfc 2961  df-sn 4560

This theorem is referenced by:  eusn  4658  uniintsn  4904  args  5950  opabiotadm  6738  mapsnd  8442