Theorem nfeuv 2032

Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2033 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 1516	. . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝑧
3	1, 2	nfbi 1577	. . . 4 ⊢ Ⅎ𝑥(𝜑 ↔ 𝑦 = 𝑧)
4	3	nfal 1564	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 ↔ 𝑦 = 𝑧)
5	4	nfex 1625	. 2 ⊢ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧)
6		df-eu 2017	. . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧))
7	6	nfbii 1461	. 2 ⊢ (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧))
8	5, 7	mpbir 145	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:   ↔ wb 104  ∀wal 1341  Ⅎwnf 1448  ∃wex 1480  ∃!weu 2014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-eu 2017

This theorem is referenced by:  nfeu  2033