Theorem rabeqsnd 29342

Description: Conditions for a restricted class abstraction to be a singleton, in deduction form. (Contributed by Thierry Arnoux, 2-Dec-2021.)

Hypotheses

Ref	Expression
rabeqsnd.0	⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
rabeqsnd.1	⊢ (𝜑 → 𝐵 ∈ 𝐴)
rabeqsnd.2	⊢ (𝜑 → 𝜒)
rabeqsnd.3	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝑥 = 𝐵)

Assertion

Ref	Expression
rabeqsnd	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵})

Distinct variable groups:   𝑥,𝐵   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rabeqsnd

Step	Hyp	Ref	Expression
1		rabeqsnd.3	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝑥 = 𝐵)
2	1	expl 648	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵))
3	2	alrimiv 1855	. . . 4 ⊢ (𝜑 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵))
4		rabeqsnd.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
5		rabeqsnd.2	. . . . . . . 8 ⊢ (𝜑 → 𝜒)
6	4, 5	jca 554	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜒))
7	6	a1d 25	. . . . . 6 ⊢ (𝜑 → (𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
8	7	alrimiv 1855	. . . . 5 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
9		eleq1 2689	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
10		rabeqsnd.0	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
11	9, 10	anbi12d 747	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
12	11	pm5.74i 260	. . . . . 6 ⊢ ((𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
13	12	albii 1747	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑥(𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
14	8, 13	sylibr 224	. . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
15	3, 14	jca 554	. . 3 ⊢ (𝜑 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓))))
16		albiim 1816	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 = 𝐵) ↔ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓))))
17	15, 16	sylibr 224	. 2 ⊢ (𝜑 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 = 𝐵))
18		rabeqsn 4214	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵} ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 = 𝐵))
19	17, 18	sylibr 224	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {crab 2916  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921  df-sn 4178

This theorem is referenced by:  repr0  30689