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Theorem abeq2d 2763
Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2761). (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 2716 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2639 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3syl6bb 276 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  {cab 2637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-12 2087  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647
This theorem is referenced by:  abeq2i  2764  fvelimab  6292  nosupbnd2  31987  ispridlc  33999  ac6s6  34110  dib1dim  36771  mapsnend  39705
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