Theorem moeu 2610

Description: Uniqueness is equivalent to existence implying unique existence. Alternate definition of the at-most-one quantifier, in terms of the existential quantifier and the unique existential quantifier. (Contributed by NM, 8-Mar-1995.) This used to be the definition of the at-most-one quantifier, while df-mo 2566 was then proved as dfmo2 2623. (Revised by BJ, 30-Sep-2022.)

Assertion

Ref	Expression
moeu	⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Proof of Theorem moeu

Step	Hyp	Ref	Expression
1		moabs 2570	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2		exmoeub 2607	. . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
3	2	pm5.74i 274	. 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
4	1, 3	bitri 278	1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812  ∃*wmo 2564  ∃!weu 2595

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2566  df-eu 2596

This theorem is used by:  dfeu  2622  dfmo2  2623  sb8mo  2628  2euexv  2658  2euex  2668  2eu1  2677  2eu1v  2678  rmo5  3385  funeu  6562  dffun8  6565  modom  9224  climmo  15646  rmoxfrd  32969  nmotru  37029  bj-moeub  37594  wl-sb8mot  38345  wl-sb8motv  38346  nexmo1  38999  moeu2  39120  moxfr  43539  funressneu  47937  funressndmafv2rn  48113