Theorem iota4

Description: Theorem *14.22 in WhiteheadRussell p. 190. (Contributed by Andrew Salmon, 12-Jul-2011)

	Ref	Expression
Assertion	iota4	⊢ ( ∃! 𝑥 𝜑 → [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		eu6	⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) )
2		biimpr	⊢ ( ( 𝜑 ↔ 𝑥 = 𝑧 ) → ( 𝑥 = 𝑧 → 𝜑 ) )
3	2	alimi	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) )
4		sb6	⊢ ( [ 𝑧 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑧 → 𝜑 ) )
5	3 4	sylibr	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → [ 𝑧 / 𝑥 ] 𝜑 )
6		iotaval	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → ( ℩ 𝑥 𝜑 ) = 𝑧 )
7	6	eqcomd	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → 𝑧 = ( ℩ 𝑥 𝜑 ) )
8		dfsbcq2	⊢ ( 𝑧 = ( ℩ 𝑥 𝜑 ) → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 ) )
9	7 8	syl	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → ( [ 𝑧 / 𝑥 ] 𝜑 ↔ [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 ) )
10	5 9	mpbid	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 )
11	10	exlimiv	⊢ ( ∃ 𝑧 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑧 ) → [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 )
12	1 11	sylbi	⊢ ( ∃! 𝑥 𝜑 → [ ( ℩ 𝑥 𝜑 ) / 𝑥 ] 𝜑 )