Theorem eu3v 2564

Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Replace a nonfreeness hypothesis with a disjoint variable condition on 𝜑, 𝑦 to reduce axiom usage. (Revised by Wolf Lammen, 29-May-2019.)

Assertion

Ref	Expression
eu3v	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eu3v

Step	Hyp	Ref	Expression
1		df-eu 2563	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		df-mo 2534	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
3	2	anbi2i 623	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
4	1, 3	bitri 274	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539  ∃wex 1781  ∃*wmo 2532  ∃!weu 2562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-mo 2534  df-eu 2563

This theorem is referenced by:  eu6  2568  eu6im  2569  euequ  2591  euae  2655  eqeu  3702  reu3  3723