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Theorem elabd 3631
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 3626 . . 3 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜒))
52, 4syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜒))
61, 5mpbird 256 1 (𝜑𝐴 ∈ {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815
This theorem is referenced by:  intidg  5412  wfrlem15OLD  8261  lubval  18205  glbval  18218  sursubmefmnd  18666  injsubmefmnd  18667  nosupfv  27006  branmfn  30876  orvcval  32869  sticksstones3  40494  rngunsnply  41409  hoidmvlelem1  44737  cfsetsnfsetf  45193
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