Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  absnsb Structured version   Visualization version   GIF version

Theorem absnsb 44521
Description: If the class abstraction {𝑥𝜑} associated with the wff 𝜑 is a singleton, the wff is true for the singleton element. (Contributed by AV, 24-Aug-2022.)
Assertion
Ref Expression
absnsb ({𝑥𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem absnsb
StepHypRef Expression
1 abid 2719 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
2 velsn 4577 . . . . 5 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
31, 2bibi12i 340 . . . 4 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
4 biimpr 219 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
53, 4sylbi 216 . . 3 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) → (𝑥 = 𝑦𝜑))
65alimi 1814 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) → ∀𝑥(𝑥 = 𝑦𝜑))
7 nfab1 2909 . . 3 𝑥{𝑥𝜑}
8 nfcv 2907 . . 3 𝑥{𝑦}
97, 8cleqf 2938 . 2 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}))
10 sb6 2088 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
116, 9, 103imtr4i 292 1 ({𝑥𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537   = wceq 1539  [wsb 2067  wcel 2106  {cab 2715  {csn 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434  df-sn 4562
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator