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Theorem abeq2d 2927
Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2925). (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 2878 . 2 (𝜑 → (𝑥𝐴𝑥 ∈ {𝑥𝜓}))
3 abid 2783 . 2 (𝑥 ∈ {𝑥𝜓} ↔ 𝜓)
42, 3syl6bb 290 1 (𝜑 → (𝑥𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2112  {cab 2779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873
This theorem is referenced by:  abeq2i  2928  fvelimab  6716  mapsnend  8575  nosupbnd2  33330  fvineqsneu  34829  fvineqsneq  34830  ispridlc  35507  ac6s6  35609  dib1dim  38460  prprspr2  44028
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