Theorem nfeu 1916

Description: Bound-variable hypothesis builder for existential uniqueness. Note that x and y needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 23-May-2018.)

Hypothesis

Ref	Expression
nfeu.1	⊢ Ⅎxφ

Assertion

Ref	Expression
nfeu	⊢ Ⅎx∃!yφ

Proof of Theorem nfeu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1418	. . 3 ⊢ Ⅎzφ
2	1	sb8eu 1910	. 2 ⊢ (∃!yφ ↔ ∃!z[z / y]φ)
3		nfeu.1	. . . 4 ⊢ Ⅎxφ
4	3	nfsb 1819	. . 3 ⊢ Ⅎx[z / y]φ
5	4	nfeuv 1915	. 2 ⊢ Ⅎx∃!z[z / y]φ
6	2, 5	nfxfr 1360	1 ⊢ Ⅎx∃!yφ

Syntax hints:  Ⅎwnf 1346  [wsb 1642  ∃!weu 1897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900

This theorem is referenced by:  hbeu  1918  eusv2nf  4154