Theorem eu3 1946

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 1412	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3	2	eu3h 1945	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381  ∃!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903

This theorem is referenced by:  eqeu  2711  reu3  2731  eunex  4285