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		nfv	⊢ Ⅎ 𝑤 ( 𝜑 ↔ 𝑥 = 𝑧 )
3	2	sb8v	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ∀ 𝑤 [ 𝑤 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑧 ) )
4		equsb3	⊢ ( [ 𝑤 / 𝑥 ] 𝑥 = 𝑧 ↔ 𝑤 = 𝑧 )
5	4	sblbis	⊢ ( [ 𝑤 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 ) )
6	5	albii	⊢ ( ∀ 𝑤 [ 𝑤 / 𝑥 ] ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ∀ 𝑤 ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 ) )
7		nfv	⊢ Ⅎ 𝑦 𝑤 = 𝑧
8	1 7	nfbi	⊢ Ⅎ 𝑦 ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 )
9		nfv	⊢ Ⅎ 𝑤 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 )
10		sbequ	⊢ ( 𝑤 = 𝑦 → ( [ 𝑤 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
11		equequ1	⊢ ( 𝑤 = 𝑦 → ( 𝑤 = 𝑧 ↔ 𝑦 = 𝑧 ) )
12	10 11	bibi12d	⊢ ( 𝑤 = 𝑦 → ( ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) ) )
13	8 9 12	cbvalv1	⊢ ( ∀ 𝑤 ( [ 𝑤 / 𝑥 ] 𝜑 ↔ 𝑤 = 𝑧 ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) )
14	3 6 13	3bitri	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) )
15	14	exbii	⊢ ( ∃ 𝑧 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) ↔ ∃ 𝑧 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) )
16		eu6	⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) )
17		eu6	⊢ ( ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑧 ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝑦 = 𝑧 ) )
18	15 16 17	3bitr4i	⊢ ( ∃! 𝑥 𝜑 ↔ ∃! 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )