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Theorem rabeqsnd 4674
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 → (𝜓𝜒))
rabeqsnd.1 (𝜑𝐵𝐴)
rabeqsnd.2 (𝜑𝜒)
rabeqsnd.3 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥 = 𝐵)
Assertion
Ref Expression
rabeqsnd (𝜑 → {𝑥𝐴𝜓} = {𝐵})
Distinct variable groups:   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem rabeqsnd
StepHypRef Expression
1 rabeqsnd.3 . . . . . 6 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝑥 = 𝐵)
21expl 457 . . . . 5 (𝜑 → ((𝑥𝐴𝜓) → 𝑥 = 𝐵))
32alrimiv 1925 . . . 4 (𝜑 → ∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝐵))
4 rabeqsnd.1 . . . . . . . 8 (𝜑𝐵𝐴)
5 rabeqsnd.2 . . . . . . . 8 (𝜑𝜒)
64, 5jca 511 . . . . . . 7 (𝜑 → (𝐵𝐴𝜒))
76a1d 25 . . . . . 6 (𝜑 → (𝑥 = 𝐵 → (𝐵𝐴𝜒)))
87alrimiv 1925 . . . . 5 (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝐵𝐴𝜒)))
9 eleq1 2827 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
10 rabeqsnd.0 . . . . . . . 8 (𝑥 = 𝐵 → (𝜓𝜒))
119, 10anbi12d 632 . . . . . . 7 (𝑥 = 𝐵 → ((𝑥𝐴𝜓) ↔ (𝐵𝐴𝜒)))
1211pm5.74i 271 . . . . . 6 ((𝑥 = 𝐵 → (𝑥𝐴𝜓)) ↔ (𝑥 = 𝐵 → (𝐵𝐴𝜒)))
1312albii 1816 . . . . 5 (∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓)) ↔ ∀𝑥(𝑥 = 𝐵 → (𝐵𝐴𝜒)))
148, 13sylibr 234 . . . 4 (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓)))
153, 14jca 511 . . 3 (𝜑 → (∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓))))
16 albiim 1887 . . 3 (∀𝑥((𝑥𝐴𝜓) ↔ 𝑥 = 𝐵) ↔ (∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓))))
1715, 16sylibr 234 . 2 (𝜑 → ∀𝑥((𝑥𝐴𝜓) ↔ 𝑥 = 𝐵))
18 rabeqsn 4672 . 2 ({𝑥𝐴𝜓} = {𝐵} ↔ ∀𝑥((𝑥𝐴𝜓) ↔ 𝑥 = 𝐵))
1917, 18sylibr 234 1 (𝜑 → {𝑥𝐴𝜓} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2106  {crab 3433  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-sn 4632
This theorem is referenced by:  rngqiprngimf1  21328  ply1dg1rt  33584  repr0  34605
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