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Theorem rabeqsnd 29680
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 445 . . . . 5 (𝜑 → ((𝑥𝐴𝜓) → 𝑥 = 𝐵))
32alrimiv 2007 . . . 4 (𝜑 → ∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝐵))
4 rabeqsnd.1 . . . . . . . 8 (𝜑𝐵𝐴)
5 rabeqsnd.2 . . . . . . . 8 (𝜑𝜒)
64, 5jca 501 . . . . . . 7 (𝜑 → (𝐵𝐴𝜒))
76a1d 25 . . . . . 6 (𝜑 → (𝑥 = 𝐵 → (𝐵𝐴𝜒)))
87alrimiv 2007 . . . . 5 (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝐵𝐴𝜒)))
9 eleq1 2838 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
10 rabeqsnd.0 . . . . . . . 8 (𝑥 = 𝐵 → (𝜓𝜒))
119, 10anbi12d 616 . . . . . . 7 (𝑥 = 𝐵 → ((𝑥𝐴𝜓) ↔ (𝐵𝐴𝜒)))
1211pm5.74i 260 . . . . . 6 ((𝑥 = 𝐵 → (𝑥𝐴𝜓)) ↔ (𝑥 = 𝐵 → (𝐵𝐴𝜒)))
1312albii 1895 . . . . 5 (∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓)) ↔ ∀𝑥(𝑥 = 𝐵 → (𝐵𝐴𝜒)))
148, 13sylibr 224 . . . 4 (𝜑 → ∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓)))
153, 14jca 501 . . 3 (𝜑 → (∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓))))
16 albiim 1968 . . 3 (∀𝑥((𝑥𝐴𝜓) ↔ 𝑥 = 𝐵) ↔ (∀𝑥((𝑥𝐴𝜓) → 𝑥 = 𝐵) ∧ ∀𝑥(𝑥 = 𝐵 → (𝑥𝐴𝜓))))
1715, 16sylibr 224 . 2 (𝜑 → ∀𝑥((𝑥𝐴𝜓) ↔ 𝑥 = 𝐵))
18 rabeqsn 4353 . 2 ({𝑥𝐴𝜓} = {𝐵} ↔ ∀𝑥((𝑥𝐴𝜓) ↔ 𝑥 = 𝐵))
1917, 18sylibr 224 1 (𝜑 → {𝑥𝐴𝜓} = {𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wcel 2145  {crab 3065  {csn 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-rab 3070  df-sn 4318
This theorem is referenced by:  repr0  31029
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