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Theorem ralsngf 4584
Description: Restricted universal quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by AV, 3-Apr-2023.)
Hypotheses
Ref Expression
rexsngf.1 𝑥𝜓
rexsngf.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ralsngf (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem ralsngf
StepHypRef Expression
1 ralsnsg 4581 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
2 rexsngf.1 . . 3 𝑥𝜓
3 rexsngf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3sbciegf 3786 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
51, 4bitrd 282 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wnf 1785  wcel 2115  wral 3126  [wsbc 3749  {csn 4540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-ral 3131  df-v 3473  df-sbc 3750  df-sn 4541
This theorem is referenced by:  reusngf  4585  ralsng  4586  rexreusng  4590  ralprgf  4603
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