Theorem bj-moeub 34173

Description: Uniqueness is equivalent to existence being equivalent to unique existence. (Contributed by BJ, 14-Oct-2022.)

Assertion

Ref	Expression
bj-moeub	⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))

Proof of Theorem bj-moeub

Step	Hyp	Ref	Expression
1		moeu 2668	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
2		euex 2662	. . . 4 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑)
3		impbi 210	. . . 4 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → ((∃!𝑥𝜑 → ∃𝑥𝜑) → (∃𝑥𝜑 ↔ ∃!𝑥𝜑)))
4	2, 3	mpi 20	. . 3 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) → (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
5		biimp 217	. . 3 ⊢ ((∃𝑥𝜑 ↔ ∃!𝑥𝜑) → (∃𝑥𝜑 → ∃!𝑥𝜑))
6	4, 5	impbii 211	. 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))
7	1, 6	bitri 277	1 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 ↔ ∃!𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∃wex 1780  ∃*wmo 2620  ∃!weu 2653

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 1970

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-mo 2622  df-eu 2654

This theorem is referenced by: (None)