Theorem eu3 2043

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.)

Hypothesis

Ref	Expression
eu3.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
eu3	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu3

Step	Hyp	Ref	Expression
1		eu3.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfri 1499	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	eu3h 2042	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468  ∃!weu 1997

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000

This theorem is referenced by:  eqeu  2849  reu3  2869  eunex  4471