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Theorem abeq2d 2288
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 2245 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2163 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3bitrdi 196 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2146  {cab 2161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-ext 2157
This theorem depends on definitions:  df-bi 117  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171
This theorem is referenced by:  fvelimab  5564  frecsuclem  6397
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