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Theorem absnsb 43256
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 2803 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
2 velsn 4576 . . . . 5 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
31, 2bibi12i 342 . . . 4 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
4 biimpr 222 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
53, 4sylbi 219 . . 3 ((𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) → (𝑥 = 𝑦𝜑))
65alimi 1808 . 2 (∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}) → ∀𝑥(𝑥 = 𝑦𝜑))
7 nfab1 2979 . . 3 𝑥{𝑥𝜑}
8 nfcv 2977 . . 3 𝑥{𝑦}
97, 8cleqf 3010 . 2 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝑥 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑦}))
10 sb6 2089 . 2 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
116, 9, 103imtr4i 294 1 ({𝑥𝜑} = {𝑦} → [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1531   = wceq 1533  [wsb 2065  wcel 2110  {cab 2799  {csn 4560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sn 4561
This theorem is referenced by: (None)
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