Theorem dfeumo 2563

Description: An elementary proof showing the reverse direction of dfmoeu 2562. Here the characterizing expression of existential uniqueness (eu6 2601) is derived from that of uniqueness (df-mo 2566). (Contributed by Wolf Lammen, 3-Oct-2023.)

Assertion

Ref	Expression
dfeumo	⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfeumo

Step	Hyp	Ref	Expression
1		ax6ev 1989	. . . . 5 ⊢ ∃𝑥 𝑥 = 𝑦
2		biimpr 222	. . . . . 6 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑))
3	2	aleximi 1852	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥𝜑))
4	1, 3	mpi 20	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑥𝜑)
5	4	exlimiv 1950	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑥𝜑)
6	5	pm4.71ri 568	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
7		abai 836	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ (∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))))
8		dfmoeu 2562	. . 3 ⊢ ((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
9	8	anbi2i 632	. 2 ⊢ ((∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
10	6, 7, 9	3bitrri 300	1 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-12 2212

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1800  df-nf 1804

This theorem is referenced by: (None)