Theorem rmo5 3394

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 2575	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
2		df-rmo 3374	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rex 3069	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-reu 3375	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	3, 4	imbi12i 349	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
6	1, 2, 5	3bitr4i 302	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∃wex 1779   ∈ wcel 2104  ∃*wmo 2530  ∃!weu 2560  ∃wrex 3068  ∃!wreu 3372  ∃*wrmo 3373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-mo 2532  df-eu 2561  df-rex 3069  df-rmo 3374  df-reu 3375

This theorem is referenced by:  nrexrmo  3395  cbvrmowOLD  3409  cbvrmo  3423  2reurex  3757  rmo0  4360  rmosn  4724  ddemeas  33530  iccpartdisj  46405