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Theorem abeq2d 2721
 Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2719). (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
abeq2d.1 (𝜑𝐴 = {𝑥𝜓})
Assertion
Ref Expression
abeq2d (𝜑 → (𝑥𝐴𝜓))

Proof of Theorem abeq2d
StepHypRef Expression
1 abeq2d.1 . . 3 (𝜑𝐴 = {𝑥𝜓})
21eleq2d 2673 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2598 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3syl6bb 275 1 (𝜑 → (𝑥𝐴𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606 This theorem is referenced by:  abeq2i  2722  fvelimab  6163  ispridlc  33039  ac6s6  33150  dib1dim  35472  mapsnend  38386
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