Theorem eu3 2065

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 1512	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	eu3h 2064	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  Ⅎwnf 1453  ∃wex 1485  ∃!weu 2019

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022

This theorem is referenced by:  eqeu  2900  reu3  2920  eunex  4545