Theorem sbmo

Description: Substitution into an at-most-one quantifier. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Assertion	sbmo	⊢ ( [ 𝑦 / 𝑥 ] ∃* 𝑧 𝜑 ↔ ∃* 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sbex	⊢ ( [ 𝑦 / 𝑥 ] ∃ 𝑤 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑤 ) ↔ ∃ 𝑤 [ 𝑦 / 𝑥 ] ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑤 ) )
2		nfv	⊢ Ⅎ 𝑥 𝑧 = 𝑤
3	2	sblim	⊢ ( [ 𝑦 / 𝑥 ] ( 𝜑 → 𝑧 = 𝑤 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑧 = 𝑤 ) )
4	3	sbalv	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑤 ) ↔ ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑧 = 𝑤 ) )
5	4	exbii	⊢ ( ∃ 𝑤 [ 𝑦 / 𝑥 ] ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑤 ) ↔ ∃ 𝑤 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑧 = 𝑤 ) )
6	1 5	bitri	⊢ ( [ 𝑦 / 𝑥 ] ∃ 𝑤 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑤 ) ↔ ∃ 𝑤 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑧 = 𝑤 ) )
7		dfmo	⊢ ( ∃* 𝑧 𝜑 ↔ ∃ 𝑤 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑤 ) )
8	7	sbbii	⊢ ( [ 𝑦 / 𝑥 ] ∃* 𝑧 𝜑 ↔ [ 𝑦 / 𝑥 ] ∃ 𝑤 ∀ 𝑧 ( 𝜑 → 𝑧 = 𝑤 ) )
9		dfmo	⊢ ( ∃* 𝑧 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑤 ∀ 𝑧 ( [ 𝑦 / 𝑥 ] 𝜑 → 𝑧 = 𝑤 ) )
10	6 8 9	3bitr4i	⊢ ( [ 𝑦 / 𝑥 ] ∃* 𝑧 𝜑 ↔ ∃* 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )