Theorem nfeuv 2044

Description: Bound-variable hypothesis builder for existential uniqueness. This is similar to nfeu 2045 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 1528	. . . . 5 ⊢ Ⅎ𝑥 𝑦 = 𝑧
3	1, 2	nfbi 1589	. . . 4 ⊢ Ⅎ𝑥(𝜑 ↔ 𝑦 = 𝑧)
4	3	nfal 1576	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 ↔ 𝑦 = 𝑧)
5	4	nfex 1637	. 2 ⊢ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧)
6		df-eu 2029	. . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧))
7	6	nfbii 1473	. 2 ⊢ (Ⅎ𝑥∃!𝑦𝜑 ↔ Ⅎ𝑥∃𝑧∀𝑦(𝜑 ↔ 𝑦 = 𝑧))
8	5, 7	mpbir 146	1 ⊢ Ⅎ𝑥∃!𝑦𝜑

Syntax hints:   ↔ wb 105  ∀wal 1351  Ⅎwnf 1460  ∃wex 1492  ∃!weu 2026

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-eu 2029

This theorem is referenced by:  nfeu  2045