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Theorem abeq2d 2277
Description: Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
abeqd.1 (𝜑𝐴 = {𝑥𝜓})
Assertion
Ref Expression
abeq2d (𝜑 → (𝑥𝐴𝜓))

Proof of Theorem abeq2d
StepHypRef Expression
1 abeqd.1 . . 3 (𝜑𝐴 = {𝑥𝜓})
21eleq2d 2234 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2152 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3bitrdi 195 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1342  wcel 2135  {cab 2150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160
This theorem is referenced by:  fvelimab  5537  frecsuclem  6366
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