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Theorem abeq2d 2166
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 2123 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2044 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3syl6bb 189 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052
This theorem is referenced by:  fvelimab  5257  frecsuclem3  6021
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