Theorem rmo5 3384

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		moeu 2609	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
2		df-rmo 3366	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rex 3088	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-reu 3367	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	3, 4	imbi12i 353	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
6	1, 2, 5	3bitr4i 306	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2563  ∃!weu 2594  ∃wrex 3087  ∃!wreu 3364  ∃*wrmo 3365

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2565  df-eu 2595  df-rex 3088  df-rmo 3366  df-reu 3367

This theorem is used by:  nrexrmo  3385  cbvrmo  3406  2reurex  3718  rmo0  4310  rmosn  4680  ddemeas  34862  iccpartdisj  48488