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Theorem abeq1i 2206
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 2083 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
2 abeqri.1 . . 3 {𝑥𝜑} = 𝐴
32eleq2i 2161 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝑥𝐴)
41, 3bitr3i 185 1 (𝜑𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1296  wcel 1445  {cab 2081
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091
This theorem is referenced by: (None)
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