Theorem euex 2227

Description: Existential uniqueness implies existence. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
euex	⊢ (∃!xφ → ∃xφ)

Proof of Theorem euex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 ⊢ Ⅎyφ
2	1	eu1 2225	. 2 ⊢ (∃!xφ ↔ ∃x(φ ∧ ∀y([y / x]φ → x = y)))
3		exsimpl 1592	. 2 ⊢ (∃x(φ ∧ ∀y([y / x]φ → x = y)) → ∃xφ)
4	2, 3	sylbi 187	1 ⊢ (∃!xφ → ∃xφ)

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648  ∃!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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208

This theorem is referenced by:  eu2  2229  exmoeu  2246  eupickbi  2270  2eu2ex  2278  2exeu  2281  euxfr  3023  fvprc  5326  tz6.12c  5348  ndmfv  5350  dff3  5421  fnoprabg  5586