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Theorem abeq2d 2944
Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2942). (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 2895 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2800 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3syl6bb 288 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {cab 2796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890
This theorem is referenced by:  abeq2i  2945  fvelimab  6730  mapsnend  8576  nosupbnd2  33113  fvineqsneu  34574  fvineqsneq  34575  ispridlc  35229  ac6s6  35331  dib1dim  38181  prprspr2  43557
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