Theorem rabeqsnd 4629

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 463	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵))
3	2	alrimiv 1959	. . . 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 521	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜒))
7	6	a1d 26	. . . . . 6 ⊢ (𝜑 → (𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
8	7	alrimiv 1959	. . . . 5 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
9		eleq1 2848	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
10		rabeqsnd.0	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
11	9, 10	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
12	11	pm5.74i 274	. . . . . 6 ⊢ ((𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
13	12	albii 1851	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑥(𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
14	8, 13	sylibr 237	. . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
15	3, 14	jca 521	. . 3 ⊢ (𝜑 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓))))
16		albiim 1921	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 = 𝐵) ↔ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓))))
17	15, 16	sylibr 237	. 2 ⊢ (𝜑 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 = 𝐵))
18		rabeqsn 4627	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵} ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 = 𝐵))
19	17, 18	sylibr 237	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {crab 3412  {csn 4583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-sn 4584

This theorem is used by:  rngqiprngimf1  21596  ply1dg1rt  34112  esplyfval0  34196  vieta  34212  repr0  35240