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Theorem ralsngf 4603
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 4598 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
2 rexsngf.1 . . 3 𝑥𝜓
3 rexsngf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3sbciegf 3806 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
51, 4bitrd 280 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wnf 1775  wcel 2105  wral 3135  [wsbc 3769  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-v 3494  df-sbc 3770  df-sn 4558
This theorem is referenced by:  reusngf  4604  ralsng  4605  rexreusng  4609  ralprgf  4622
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