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Theorem abeq2d 2132
Description: Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
abeqd.1 (φA = {xψ})
Assertion
Ref Expression
abeq2d (φ → (x Aψ))

Proof of Theorem abeq2d
StepHypRef Expression
1 abeqd.1 . . 3 (φA = {xψ})
21eleq2d 2089 . 2 (φ → (x Ax {xψ}))
3 abid 2010 . 2 (x {xψ} ↔ ψ)
42, 3syl6bb 185 1 (φ → (x Aψ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  {cab 2008
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018
This theorem is referenced by:  fvelimab  5154  frecsuclem3  5906
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