Theorem reurmo 2827

Description: Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)

Assertion

Ref	Expression
reurmo	⊢ (∃!x ∈ A φ → ∃*x ∈ A φ)

Proof of Theorem reurmo

Step	Hyp	Ref	Expression
1		reu5 2825	. 2 ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃*x ∈ A φ))
2	1	simprbi 450	1 ⊢ (∃!x ∈ A φ → ∃*x ∈ A φ)

Syntax hints:   → wi 4  ∃wrex 2616  ∃!wreu 2617  ∃*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-rex 2621  df-reu 2622  df-rmo 2623

This theorem is referenced by: (None)