Theorem moeu 2583

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 2540 was then proved as dfmo 2596. (Revised by BJ, 30-Sep-2022.)

Assertion

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

Proof of Theorem moeu

Step	Hyp	Ref	Expression
1		moabs 2543	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2		exmoeub 2580	. . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
3	2	pm5.74i 270	. 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
4	1, 3	bitri 274	1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1783  ∃*wmo 2538  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540  df-eu 2569

This theorem is referenced by:  dfeu  2595  dfmo  2596  sb8mo  2601  cbvmowOLD  2604  2euexv  2633  2euex  2643  2eu1  2652  2eu1v  2653  rmo5  3355  funeu  6443  dffun8  6446  modom  8953  climmo  15194  rmoxfrd  30742  nmotru  34524  amosym1  34542  bj-moeub  34960  wl-sb8mot  35660  nexmo1  36313  moxfr  40430  funressneu  44428  funressndmafv2rn  44602