Theorem eu2 2613

Description: An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)

Hypothesis

Ref	Expression
eu2.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
eu2	⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem eu2

Step	Hyp	Ref	Expression
1		df-eu 2573	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2		eu2.nf	. . . 4 ⊢ Ⅎ𝑦𝜑
3	2	mo3 2568	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
4	3	anbi2i 629	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
5	1, 4	bitri 276	1 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786  Ⅎwnf 1790  [wsb 2073  ∃*wmo 2541  ∃!weu 2572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573

This theorem is referenced by:  reu2  3666  bnj1321  35209