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Theorem elabd 3666
Description: Explicit demonstration the class {𝑥𝜓} is not empty by the example 𝐴. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 23-Mar-2024.)
Hypotheses
Ref Expression
elabd.1 (𝜑𝐴𝑉)
elabd.2 (𝜑𝜒)
elabd.3 (𝑥 = 𝐴 → (𝜓𝜒))
Assertion
Ref Expression
elabd (𝜑𝐴 ∈ {𝑥𝜓})
Distinct variable groups:   𝑥,𝐴   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem elabd
StepHypRef Expression
1 elabd.2 . 2 (𝜑𝜒)
2 elabd.1 . . 3 (𝜑𝐴𝑉)
3 elabd.3 . . . 4 (𝑥 = 𝐴 → (𝜓𝜒))
43elabg 3663 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜒))
52, 4syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜒))
61, 5mpbird 258 1 (𝜑𝐴 ∈ {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {cab 2796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960
This theorem is referenced by:  wfrlem15  7958  lubval  17582  glbval  17595  branmfn  29809  orvcval  31614  nosupfv  33103  rngunsnply  39651  hoidmvlelem1  42754  sursubmefmnd  43993  injsubmefmnd  43994
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