Theorem reurmo 3435

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 3432	. 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 3141  ∃!wreu 3142  ∃*wrmo 3143

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 2654  df-rex 3146  df-reu 3147  df-rmo 3148

This theorem is referenced by:  reuimrmo  3738  reuxfr1d  3743  2reurmo  3753  2rexreu  3755  2reu2  3884  enqeq  10358  eqsqrtd  14729  efgred2  18881  0frgp  18907  frgpnabllem2  18996  frgpcyg  20722  lmieu  26572  poimirlem25  34919  poimirlem26  34920