Theorem reu5 2825

Description: Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
reu5	⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃*x ∈ A φ))

Proof of Theorem reu5

Step	Hyp	Ref	Expression
1		eu5 2242	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ (∃x(x ∈ A ∧ φ) ∧ ∃*x(x ∈ A ∧ φ)))
2		df-reu 2622	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
3		df-rex 2621	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
4		df-rmo 2623	. . 3 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
5	3, 4	anbi12i 678	. 2 ⊢ ((∃x ∈ A φ ∧ ∃*x ∈ A φ) ↔ (∃x(x ∈ A ∧ φ) ∧ ∃*x(x ∈ A ∧ φ)))
6	1, 2, 5	3bitr4i 268	1 ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃*x ∈ A φ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∃wrex 2616  ∃!wreu 2617  ∃*wrmo 2618

This proof depends on 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 proof 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 used by:  reurex  2826  reurmo  2827  reu4  3031  reueq  3034  fncnv  5159