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Theorem abeq2d 2201
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 2158 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2077 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3syl6bb 195 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1290  wcel 1439  {cab 2075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085
This theorem is referenced by:  fvelimab  5373  frecsuclem  6185
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