Theorem euex 1021

Description: Existential uniqueness implies existence.

Assertion

Ref	Expression
euex	⊢ (∃!xφ → ∃xφ)

Proof of Theorem euex

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 ⊢ (φ → ∀yφ)
2	1	eu1 1019	. . 3 ⊢ (∃!xφ ↔ ∃x(φ ∧ ∀y([y / x]φ → x = y)))
3		19.40 773	. . 3 ⊢ (∃x(φ ∧ ∀y([y / x]φ → x = y)) → (∃xφ ∧ ∃x∀y([y / x]φ → x = y)))
4	2, 3	sylbi 174	. 2 ⊢ (∃!xφ → (∃xφ ∧ ∃x∀y([y / x]φ → x = y)))
5	4	pm3.26d 258	1 ⊢ (∃!xφ → ∃xφ)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  [wsb 852  ∃!weu 1007

This theorem is referenced by:  eu2 1023  exmoeu 1039  2eu2ex 1063  reurex 1337  euxfr 1436  zfrep6 2744  fnopabg 2745  tz6.12c 2846  ndmfv 2848  fnoprab 3038  aceq5lem5 3562  hlimeu 5146

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009

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