Theorem iota4an

Description: Theorem *14.23 in WhiteheadRussell p. 191. (Contributed by Andrew Salmon, 12-Jul-2011)

	Ref	Expression
Assertion	iota4an	⊢ ( ∃! 𝑥 ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		iota4	⊢ ( ∃! 𝑥 ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( 𝜑 ∧ 𝜓 ) )
2		iotaex	⊢ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ∈ V
3		simpl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
4	3	sbcth	⊢ ( ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ∈ V → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) )
5	2 4	ax-mp	⊢ [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
6		sbcimg	⊢ ( ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) ∈ V → ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) ↔ ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 ) ) )
7	2 6	ax-mp	⊢ ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( ( 𝜑 ∧ 𝜓 ) → 𝜑 ) ↔ ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 ) )
8	5 7	mpbi	⊢ ( [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 )
9	1 8	syl	⊢ ( ∃! 𝑥 ( 𝜑 ∧ 𝜓 ) → [ ( ℩ 𝑥 ( 𝜑 ∧ 𝜓 ) ) / 𝑥 ] 𝜑 )