Theorem reurmo 3340

Description: Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)

Assertion

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

Proof of Theorem reurmo

Step	Hyp	Ref	Expression
1		reu5 3337	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
2	1	simprbi 500	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∃wrex 3062  ∃!wreu 3063  ∃*wrmo 3064

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 210  df-an 400  df-eu 2568  df-rex 3067  df-reu 3068  df-rmo 3069

This theorem is referenced by:  reuimrmo  3658  reuxfr1d  3663  2reurmo  3672  2rexreu  3675  2reu2  3810  enqeq  10548  eqsqrtd  14931  efgred2  19143  0frgp  19169  frgpnabllem2  19259  frgpcyg  20538  lmieu  26875  poimirlem25  35539  poimirlem26  35540  addinvcom  40121