Theorem elabgt

Description: Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg .) (Contributed by NM, 7-Nov-2005) (Proof shortened by Andrew Salmon, 8-Jun-2011) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024)

	Ref	Expression
Assertion	elabgt	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		elab6g	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
2	1	adantr	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
3		elisset	⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 = 𝐴 )
4		biimp	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) )
5	4	imim3i	⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ( 𝑥 = 𝐴 → 𝜑 ) → ( 𝑥 = 𝐴 → 𝜓 ) ) )
6	5	al2imi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) )
7		19.23v	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
8	6 7	syl6ib	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) )
9	8	com3r	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) → 𝜓 ) ) )
10		biimpr	⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜓 → 𝜑 ) )
11	10	imim2i	⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) )
12	11	alimi	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) )
13		bi2.04	⊢ ( ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) ↔ ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
14	13	albii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) ) )
15		19.21v	⊢ ( ∀ 𝑥 ( 𝜓 → ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
16	14 15	sylbb	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜑 ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
17	12 16	syl	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
18	17	a1i	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ) )
19	9 18	impbidd	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) ) )
20	3 19	syl	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) ) )
21	20	imp	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
22	2 21	bitrd	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ) → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )