Theorem nfeu 2098

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 1576	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	sb8eu 2092	. 2 ⊢ (∃!𝑦𝜑 ↔ ∃!𝑧[𝑧 / 𝑦]𝜑)
3		nfeu.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4	3	nfsb 1999	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜑
5	4	nfeuv 2097	. 2 ⊢ Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜑
6	2, 5	nfxfr 1522	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:  Ⅎwnf 1508  [wsb 1810  ∃!weu 2079

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082

This theorem is referenced by:  hbeu  2100  eusv2nf  4553