Theorem pm14.12

Description: Theorem *14.12 in WhiteheadRussell p. 184. (Contributed by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Assertion	pm14.12	⊢ ( ∃! 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		eumo	⊢ ( ∃! 𝑥 𝜑 → ∃* 𝑥 𝜑 )
2		nfv	⊢ Ⅎ 𝑦 𝜑
3	2	mo3	⊢ ( ∃* 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
4		sbsbc	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 )
5	4	anbi2i	⊢ ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) )
6	5	imbi1i	⊢ ( ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ↔ ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
7	6	2albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
8	3 7	bitri	⊢ ( ∃* 𝑥 𝜑 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )
9	1 8	sylib	⊢ ( ∃! 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 ( ( 𝜑 ∧ [ 𝑦 / 𝑥 ] 𝜑 ) → 𝑥 = 𝑦 ) )