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Theorem ralsngf 4670
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 4667 . 2 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
2 rexsngf.1 . . 3 𝑥𝜓
3 rexsngf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3sbciegf 3811 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
51, 4bitrd 279 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wnf 1777  wcel 2098  wral 3055  [wsbc 3772  {csn 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-v 3470  df-sbc 3773  df-sn 4624
This theorem is referenced by:  reusngf  4671  ralsngOLD  4676  rexreusng  4678  ralprgf  4691
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