Theorem rmo5 2828

Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)

Assertion

Ref	Expression
rmo5	⊢ (∃*x ∈ A φ ↔ (∃x ∈ A φ → ∃!x ∈ A φ))

Proof of Theorem rmo5

Step	Hyp	Ref	Expression
1		df-mo 2209	. 2 ⊢ (∃*x(x ∈ A ∧ φ) ↔ (∃x(x ∈ A ∧ φ) → ∃!x(x ∈ A ∧ φ)))
2		df-rmo 2623	. 2 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
3		df-rex 2621	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
4		df-reu 2622	. . 3 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
5	3, 4	imbi12i 316	. 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))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∃wrex 2616  ∃!wreu 2617  ∃*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-mo 2209  df-rex 2621  df-reu 2622  df-rmo 2623

This theorem is referenced by:  nrexrmo  2829  cbvrmo  2835