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Theorem rabeqsnd 4670
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 456 . . . . 5 (𝜑 → ((𝑥𝐴𝜓) → 𝑥 = 𝐵))
32alrimiv 1928 . . . 4 (𝜑 → ∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝐵))
4 rabeqsnd.1 . . . . . . . 8 (𝜑𝐵𝐴)
5 rabeqsnd.2 . . . . . . . 8 (𝜑𝜒)
64, 5jca 510 . . . . . . 7 (𝜑 → (𝐵𝐴𝜒))
76a1d 25 . . . . . 6 (𝜑 → (𝑥 = 𝐵 → (𝐵𝐴𝜒)))
87alrimiv 1928 . . . . 5 (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝐵𝐴𝜒)))
9 eleq1 2819 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
10 rabeqsnd.0 . . . . . . . 8 (𝑥 = 𝐵 → (𝜓𝜒))
119, 10anbi12d 629 . . . . . . 7 (𝑥 = 𝐵 → ((𝑥𝐴𝜓) ↔ (𝐵𝐴𝜒)))
1211pm5.74i 270 . . . . . 6 ((𝑥 = 𝐵 → (𝑥𝐴𝜓)) ↔ (𝑥 = 𝐵 → (𝐵𝐴𝜒)))
1312albii 1819 . . . . 5 (∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓)) ↔ ∀𝑥(𝑥 = 𝐵 → (𝐵𝐴𝜒)))
148, 13sylibr 233 . . . 4 (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓)))
153, 14jca 510 . . 3 (𝜑 → (∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓))))
16 albiim 1890 . . 3 (∀𝑥((𝑥𝐴𝜓) ↔ 𝑥 = 𝐵) ↔ (∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓))))
1715, 16sylibr 233 . 2 (𝜑 → ∀𝑥((𝑥𝐴𝜓) ↔ 𝑥 = 𝐵))
18 rabeqsn 4668 . 2 ({𝑥𝐴𝜓} = {𝐵} ↔ ∀𝑥((𝑥𝐴𝜓) ↔ 𝑥 = 𝐵))
1917, 18sylibr 233 1 (𝜑 → {𝑥𝐴𝜓} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1537   = wceq 1539  wcel 2104  {crab 3430  {csn 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-sn 4628
This theorem is referenced by:  rngqiprngimf1  21059  repr0  33921
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