Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralsnsg Structured version   Visualization version   GIF version

Theorem ralsnsg 4568
 Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
ralsnsg (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem ralsnsg
StepHypRef Expression
1 df-ral 3111 . . 3 (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝜑))
2 velsn 4541 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
32imbi1i 353 . . . 4 ((𝑥 ∈ {𝐴} → 𝜑) ↔ (𝑥 = 𝐴𝜑))
43albii 1821 . . 3 (∀𝑥(𝑥 ∈ {𝐴} → 𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑))
51, 4bitri 278 . 2 (∀𝑥 ∈ {𝐴}𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑))
6 sbc6g 3749 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
75, 6bitr4id 293 1 (𝐴𝑉 → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106  [wsbc 3720  {csn 4525 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-sbc 3721  df-sn 4526 This theorem is referenced by:  ralsngf  4571  ixpsnval  8449  ac6sfi  8748  rexfiuz  14701  prmind2  16021  finixpnum  35058
 Copyright terms: Public domain W3C validator