Theorem nfeu 2057

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 1539	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	sb8eu 2051	. 2 ⊢ (∃!𝑦𝜑 ↔ ∃!𝑧[𝑧 / 𝑦]𝜑)
3		nfeu.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4	3	nfsb 1958	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜑
5	4	nfeuv 2056	. 2 ⊢ Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜑
6	2, 5	nfxfr 1485	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:  Ⅎwnf 1471  [wsb 1773  ∃!weu 2038

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041

This theorem is referenced by:  hbeu  2059  eusv2nf  4471