Theorem euex 2056

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 1526	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	eu1 2051	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
3		exsimpl 1617	. 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 1351  ∃wex 1492  [wsb 1762  ∃!weu 2026

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029

This theorem is referenced by:  eu2  2070  eu3h  2071  eu5  2073  exmoeudc  2089  eupickbi  2108  2eu2ex  2115  euxfrdc  2925  repizf  4121  eusvnf  4455  eusvnfb  4456  tz6.12c  5547  ndmfvg  5548  nfvres  5550  0fv  5552  eusvobj2  5863  fnoprabg  5978  0g0  12800  txcn  13860