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Theorem absnsb 44082
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 2720 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
2 velsn 4532 . . . . 5 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
31, 2bibi12i 343 . . . 4 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
4 biimpr 223 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
53, 4sylbi 220 . . 3 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) → (𝑥 = 𝑦𝜑))
65alimi 1818 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) → ∀𝑥(𝑥 = 𝑦𝜑))
7 nfab1 2901 . . 3 𝑥{𝑥𝜑}
8 nfcv 2899 . . 3 𝑥{𝑦}
97, 8cleqf 2930 . 2 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}))
10 sb6 2095 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
116, 9, 103imtr4i 295 1 ({𝑥𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1540   = wceq 1542  [wsb 2074  wcel 2114  {cab 2716  {csn 4516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3400  df-sn 4517
This theorem is referenced by: (None)
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