Theorem reurmo 3300

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 3298	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
2	1	simprbi 483	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∃wrex 3051  ∃!wreu 3052  ∃*wrmo 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-eu 2611  df-mo 2612  df-rex 3056  df-reu 3057  df-rmo 3058

This theorem is referenced by:  reuxfrd  5042  enqeq  9948  eqsqrtd  14306  efgred2  18366  0frgp  18392  frgpnabllem2  18477  frgpcyg  20124  lmieu  25875  reuxfr4d  29638  poimirlem25  33747  poimirlem26  33748  reuimrmo  41684  2reurmo  41688  2rexreu  41691  2reu2  41693