Theorem nfeu 1967

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 1466	. . 3 ⊢ Ⅎ𝑧𝜑
2	1	sb8eu 1961	. 2 ⊢ (∃!𝑦𝜑 ↔ ∃!𝑧[𝑧 / 𝑦]𝜑)
3		nfeu.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4	3	nfsb 1870	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]𝜑
5	4	nfeuv 1966	. 2 ⊢ Ⅎ𝑥∃!𝑧[𝑧 / 𝑦]𝜑
6	2, 5	nfxfr 1408	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:  Ⅎwnf 1394  [wsb 1692  ∃!weu 1948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951

This theorem is referenced by:  hbeu  1969  eusv2nf  4278