Theorem reurmo 3436

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 3433	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
2	1	simprbi 499	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∃wrex 3142  ∃!wreu 3143  ∃*wrmo 3144

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 209  df-an 399  df-eu 2653  df-rex 3147  df-reu 3148  df-rmo 3149

This theorem is referenced by:  reuimrmo  3739  reuxfr1d  3744  2reurmo  3754  2rexreu  3756  2reu2  3885  enqeq  10359  eqsqrtd  14730  efgred2  18882  0frgp  18908  frgpnabllem2  18997  frgpcyg  20723  lmieu  26573  poimirlem25  34921  poimirlem26  34922