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Metamath Proof Explorer

Theorem sb8eulem

Description: Lemma. Factor out the common proof skeleton of sb8euv and sb8eu . Variable substitution in unique existential quantifier. (Contributed by NM, 7-Aug-1994) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 24-Aug-2019) Factor out common proof lines. (Revised by Wolf Lammen, 9-Feb-2023)

Ref Expression
Hypothesis sb8eulem.nfsb ⊢ Ⅎ 𝑦 [ 𝑤 / 𝑥 ] 𝜑
Assertion sb8eulem ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 sb8eulem.nfsb ⊢ Ⅎ 𝑦 [ 𝑤 / 𝑥 ] 𝜑
2 sb8v ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ∀ 𝑤 [ 𝑤 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑧 ) )
3 equsb3 ⊢ ( [ 𝑤 / 𝑥 ] 𝑥 = 𝑧 ↔ 𝑤 = 𝑧 )
4 3 sblbis ⊢ ( [ 𝑤 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 ) )
5 4 albii ⊢ ( ∀ 𝑤 [ 𝑤 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ∀ 𝑤 ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 ) )
6 nfv ⊢ Ⅎ 𝑦 𝑤 = 𝑧
7 1 6 nfbi ⊢ Ⅎ 𝑦 ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 )
8 nfv ⊢ Ⅎ 𝑤 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 )
9 sbequ ⊢ ( 𝑤 = 𝑦 → ( [ 𝑤 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
10 equequ1 ⊢ ( 𝑤 = 𝑦 → ( 𝑤 = 𝑧 ↔ 𝑦 = 𝑧 ) )
11 9 10 bibi12d ⊢ ( 𝑤 = 𝑦 → ( ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) ) )
12 7 8 11 cbvalv1 ⊢ ( ∀ 𝑤 ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) )
13 2 5 12 3bitri ⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) )
14 13 exbii ⊢ ( ∃ 𝑧 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ∃ 𝑧 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) )
15 eu6 ⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) )
16 eu6 ⊢ ( ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑧 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) )
17 14 15 16 3bitr4i ⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )