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Theorem absnsb 46942
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 2721 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
2 velsn 4664 . . . . 5 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
31, 2bibi12i 339 . . . 4 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
4 biimpr 220 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
53, 4sylbi 217 . . 3 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) → (𝑥 = 𝑦𝜑))
65alimi 1809 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) → ∀𝑥(𝑥 = 𝑦𝜑))
7 nfab1 2910 . . 3 𝑥{𝑥𝜑}
8 nfcv 2908 . . 3 𝑥{𝑦}
97, 8cleqf 2940 . 2 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}))
10 sb6 2085 . 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 206  wal 1535   = wceq 1537  [wsb 2064  wcel 2108  {cab 2717  {csn 4648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-v 3490  df-sn 4649
This theorem is referenced by: (None)
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