Theorem moeu 2578

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 2535 was then proved as dfmo 2591. (Revised by BJ, 30-Sep-2022.)

Assertion

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

Proof of Theorem moeu

Step	Hyp	Ref	Expression
1		moabs 2538	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
2		exmoeub 2575	. . 3 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 ↔ ∃!𝑥𝜑))
3	2	pm5.74i 271	. 2 ⊢ ((∃𝑥𝜑 → ∃*𝑥𝜑) ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
4	1, 3	bitri 275	1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1780  ∃*wmo 2533  ∃!weu 2563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-mo 2535  df-eu 2564

This theorem is referenced by:  dfeu  2590  dfmo  2591  sb8mo  2596  2euexv  2626  2euex  2636  2eu1  2646  2eu1v  2647  rmo5  3364  funeu  6501  dffun8  6504  modom  9130  climmo  15459  rmoxfrd  32464  nmotru  36442  bj-moeub  36883  wl-sb8mot  37614  wl-sb8motv  37615  nexmo1  38282  moeu2  38390  moxfr  42725  funressneu  47078  funressndmafv2rn  47254