Theorem euex 2075

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 1540	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	eu1 2070	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
3		exsimpl 1631	. 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 1362  ∃wex 1506  [wsb 1776  ∃!weu 2045

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048

This theorem is referenced by:  eu2  2089  eu3h  2090  eu5  2092  exmoeudc  2108  eupickbi  2127  2eu2ex  2134  euxfrdc  2950  repizf  4149  eusvnf  4488  eusvnfb  4489  tz6.12c  5588  ndmfvg  5589  elfvm  5591  nfvres  5592  0fv  5594  eusvobj2  5908  fnoprabg  6023  0g0  13019  txcn  14511