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Theorem abeq2d 2918
Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2916). (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 2871 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2794 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3syl6bb 278 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  wcel 2156  {cab 2792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-12 2214  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802
This theorem is referenced by:  abeq2i  2919  fvelimab  6470  mapsnend  8267  nosupbnd2  32178  ispridlc  34175  ac6s6  34285  dib1dim  36940
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