Theorem nfeuv 2104

Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2105 but has the additional condition that 𝑥 and 𝑦 must be distinct. (Contributed by Jim Kingdon, 23-May-2018.)

Hypothesis

Ref	Expression
nfeuv.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfeuv	⊢ Ⅎ𝑥∃!𝑦𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem nfeuv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfeuv.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
2		nfv 1581	. . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝑧
3	1, 2	nfbi 1642	. . . 4 ⊢ Ⅎ𝑥(𝜑 ↔ 𝑦 = 𝑧)
4	3	nfal 1629	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 ↔ 𝑦 = 𝑧)
5	4	nfex 1690	. 2 ⊢ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧)
6		df-eu 2089	. . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧))
7	6	nfbii 1526	. 2 ⊢ (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧))
8	5, 7	mpbir 146	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:   ↔ wb 105  ∀wal 1400  Ⅎwnf 1513  ∃wex 1545  ∃!weu 2086

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-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-eu 2089

This theorem is referenced by:  nfeu  2105