Theorem iotasbc5

Description: Theorem *14.205 in WhiteheadRussell p. 190. (Contributed by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Assertion	iotasbc5	⊢ ( ∃! 𝑥 𝜑 → ( [ ( ℩ 𝑥 𝜑 ) / 𝑦 ] 𝜓 ↔ ∃ 𝑦 ( 𝑦 = ( ℩ 𝑥 𝜑 ) ∧ 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sbc5	⊢ ( [ ( ℩ 𝑥 𝜑 ) / 𝑦 ] 𝜓 ↔ ∃ 𝑦 ( 𝑦 = ( ℩ 𝑥 𝜑 ) ∧ 𝜓 ) )
2	1	a1i	⊢ ( ∃! 𝑥 𝜑 → ( [ ( ℩ 𝑥 𝜑 ) / 𝑦 ] 𝜓 ↔ ∃ 𝑦 ( 𝑦 = ( ℩ 𝑥 𝜑 ) ∧ 𝜓 ) ) )