Theorem clelab

Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993) (Proof shortened by Wolf Lammen, 16-Nov-2019) Avoid ax-11 , see sbc5ALT for more details. (Revised by SN, 2-Sep-2024)

	Ref	Expression
Assertion	clelab	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		elisset	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } → ∃ 𝑦 𝑦 = 𝐴 )
2		exsimpl	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) → ∃ 𝑥 𝑥 = 𝐴 )
3		eqeq1	⊢ ( 𝑥 = 𝑦 → ( 𝑥 = 𝐴 ↔ 𝑦 = 𝐴 ) )
4	3	cbvexvw	⊢ ( ∃ 𝑥 𝑥 = 𝐴 ↔ ∃ 𝑦 𝑦 = 𝐴 )
5	2 4	sylib	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) → ∃ 𝑦 𝑦 = 𝐴 )
6		df-clab	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ [ 𝑦 / 𝑥 ] 𝜑 )
7		sb5	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )
8	6 7	bitri	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )
9		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝐴 ∈ { 𝑥 ∣ 𝜑 } ) )
10		eqeq2	⊢ ( 𝑦 = 𝐴 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝐴 ) )
11	10	anbi1d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ( 𝑥 = 𝐴 ∧ 𝜑 ) ) )
12	11	exbidv	⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ) )
13	9 12	bibi12d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ) ↔ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ) ) )
14	8 13	mpbii	⊢ ( 𝑦 = 𝐴 → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ) )
15	14	exlimiv	⊢ ( ∃ 𝑦 𝑦 = 𝐴 → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ) )
16	1 5 15	pm5.21nii	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) )