Theorem reurmo 3342

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 3340	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
2	1	simprbi 491	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∃wrex 3088  ∃!wreu 3089  ∃*wrmo 3090

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 199  df-an 386  df-eu 2607  df-rex 3093  df-reu 3094  df-rmo 3095

This theorem is referenced by:  reuxfrd  5089  enqeq  10042  eqsqrtd  14445  efgred2  18478  0frgp  18504  frgpnabllem2  18589  frgpcyg  20240  lmieu  26025  reuxfr4d  29845  poimirlem25  33915  poimirlem26  33916  reuimrmo  41943  2reurmo  41947  2rexreu  41950  2reu2  41952