Theorem reurex 2712

Description: Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)

Assertion

Ref	Expression
reurex	⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)

Proof of Theorem reurex

Step	Hyp	Ref	Expression
1		reu5 2711	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
2	1	simplbi 274	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4  ∃wrex 2473  ∃!wreu 2474  ∃*wrmo 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-rex 2478  df-reu 2479  df-rmo 2480

This theorem is referenced by:  reu3  2950  prsrriota  7848  elrealeu  7889  modprm0  12392  issrgid  13477  isringid  13521  ivthinc  14797