Theorem euex 2084

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 1549	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	eu1 2079	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑥 = 𝑦)))
3		exsimpl 1640	. 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 1371  ∃wex 1515  [wsb 1785  ∃!weu 2054

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057

This theorem is referenced by:  eu2  2098  eu3h  2099  eu5  2101  exmoeudc  2117  eupickbi  2136  2eu2ex  2143  euxfrdc  2959  repizf  4160  eusvnf  4500  eusvnfb  4501  tz6.12c  5606  ndmfvg  5607  elfvm  5609  nfvres  5610  0fv  5612  eusvobj2  5930  fnoprabg  6046  0g0  13208  txcn  14747