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Theorem abeq2d 2147
 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 2104 . 2 (φ → (x Ax {xψ}))
3 abid 2025 . 2 (x {xψ} ↔ ψ)
42, 3syl6bb 185 1 (φ → (x Aψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023 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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033 This theorem is referenced by:  fvelimab  5172  frecsuclem3  5929
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