Theorem rmo5 3383

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 2571	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
2		df-rmo 3363	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rex 3060	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
4		df-reu 3364	. . 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 1773   ∈ wcel 2098  ∃*wmo 2526  ∃!weu 2556  ∃wrex 3059  ∃!wreu 3361  ∃*wrmo 3362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-mo 2528  df-eu 2557  df-rex 3060  df-rmo 3363  df-reu 3364

This theorem is referenced by:  nrexrmo  3384  cbvrmo  3411  2reurex  3752  rmo0  4359  rmosn  4725  ddemeas  33983  iccpartdisj  46911