Theorem rabeqsnd 30749

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 457	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵))
3	2	alrimiv 1931	. . . 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 511	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ∧ 𝜒))
7	6	a1d 25	. . . . . 6 ⊢ (𝜑 → (𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
8	7	alrimiv 1931	. . . . 5 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
9		eleq1 2826	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
10		rabeqsnd.0	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
11	9, 10	anbi12d 630	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝐵 ∈ 𝐴 ∧ 𝜒)))
12	11	pm5.74i 270	. . . . . 6 ⊢ ((𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
13	12	albii 1823	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ ∀𝑥(𝑥 = 𝐵 → (𝐵 ∈ 𝐴 ∧ 𝜒)))
14	8, 13	sylibr 233	. . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
15	3, 14	jca 511	. . 3 ⊢ (𝜑 → (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓))))
16		albiim 1893	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 = 𝐵) ↔ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜓))))
17	15, 16	sylibr 233	. 2 ⊢ (𝜑 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 = 𝐵))
18		rabeqsn 4599	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵} ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝑥 = 𝐵))
19	17, 18	sylibr 233	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  {crab 3067  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-sn 4559

This theorem is referenced by:  repr0  32491