Theorem eumo0 2228

Description: Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.)

Hypothesis

Ref	Expression
eumo0.1	⊢ Ⅎyφ

Assertion

Ref	Expression
eumo0	⊢ (∃!xφ → ∃y∀x(φ → x = y))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem eumo0

Step	Hyp	Ref	Expression
1		eumo0.1	. . 3 ⊢ Ⅎyφ
2	1	euf 2210	. 2 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
3		bi1 178	. . . 4 ⊢ ((φ ↔ x = y) → (φ → x = y))
4	3	alimi 1559	. . 3 ⊢ (∀x(φ ↔ x = y) → ∀x(φ → x = y))
5	4	eximi 1576	. 2 ⊢ (∃y∀x(φ ↔ x = y) → ∃y∀x(φ → x = y))
6	2, 5	sylbi 187	1 ⊢ (∃!xφ → ∃y∀x(φ → x = y))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544  ∃!weu 2204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  eu2  2229  mo2  2233