Theorem euex 2109

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		ax-17 1574	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	eu1 2104	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
3		exsimpl 1665	. 2 ⊢ (∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)) → ∃𝑥𝜑)
4	2, 3	sylbi 121	1 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1395  ∃wex 1540  [wsb 1810  ∃!weu 2079

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-eu 2082

This theorem is referenced by:  eu2  2124  eu3h  2125  eu5  2127  exmoeudc  2143  eupickbi  2162  2eu2ex  2169  euxfrdc  2992  repizf  4205  eusvnf  4550  eusvnfb  4551  tz6.12c  5669  ndmfvg  5670  elfvm  5672  nfvres  5675  0fv  5677  eusvobj2  6003  fnoprabg  6121  0g0  13458  txcn  14998