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Theorem abeq1i 2252
 Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.)
Hypothesis
Ref Expression
abeqri.1 {𝑥𝜑} = 𝐴
Assertion
Ref Expression
abeq1i (𝜑𝑥𝐴)

Proof of Theorem abeq1i
StepHypRef Expression
1 abid 2128 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
2 abeqri.1 . . 3 {𝑥𝜑} = 𝐴
32eleq2i 2207 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝑥𝐴)
41, 3bitr3i 185 1 (𝜑𝑥𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {cab 2126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136 This theorem is referenced by: (None)
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