Theorem reurmo 3388

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 3385	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
2	1	simprbi 498	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∃wrex 3070  ∃!wreu 3328  ∃*wrmo 3329

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 206  df-an 398  df-eu 2567  df-rex 3071  df-rmo 3331  df-reu 3332

This theorem is referenced by:  reuimrmo  3685  reuxfr1d  3690  2reurmo  3699  2rexreu  3702  2reu2  3836  enqeq  10740  eqsqrtd  15128  efgred2  19408  0frgp  19434  frgpnabllem2  19524  frgpcyg  20830  lmieu  27194  poimirlem25  35850  poimirlem26  35851  addinvcom  40608