Theorem eu3 2101

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 1543	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	eu3h 2100	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371  Ⅎwnf 1484  ∃wex 1516  ∃!weu 2055

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058

This theorem is referenced by:  eqeu  2947  reu3  2967  eunex  4622