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Metamath Proof Explorer

Theorem eu1

Description: An alternate way to express uniqueness used by some authors. Exercise 2(b) of Margaris p. 110. (Contributed by NM, 20-Aug-1993) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 29-Oct-2018) Avoid ax-13 . (Revised by Wolf Lammen, 7-Feb-2023)

Ref Expression
Hypothesis eu1.nf ⊢ Ⅎ 𝑦 𝜑
Assertion eu1 ( ∃! 𝑥 𝜑 ↔ ∃ 𝑥 ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) )

Proof

Step Hyp Ref Expression
1 eu1.nf ⊢ Ⅎ 𝑦 𝜑
2 nfs1v ⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3 2 euf ⊢ ( ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑥 ) )
4 1 sb8euv ⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
5 1 sb6rfv ⊢ ( 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) )
6 equcom ⊢ ( 𝑥 = 𝑦 ↔ 𝑦 = 𝑥 )
7 6 imbi2i ⊢ ( ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 = 𝑥 ) )
8 7 albii ⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 = 𝑥 ) )
9 5 8 anbi12ci ⊢ ( ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) ↔ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 = 𝑥 ) ∧ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )
10 albiim ⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑥 ) ↔ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑦 = 𝑥 ) ∧ ∀ 𝑦 ( 𝑦 = 𝑥 → [ 𝑦 / 𝑥 ] 𝜑 ) ) )
11 9 10 bitr4i ⊢ ( ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑥 ) )
12 11 exbii ⊢ ( ∃ 𝑥 ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) ↔ ∃ 𝑥 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑥 ) )
13 3 4 12 3bitr4i ⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑥 ( 𝜑 ∧ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑥 = 𝑦 ) ) )