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Theorem abeq1i 2874
Description: Equality of a class variable and a class abstraction (inference rule). (Contributed by NM, 31-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Nov-2019.)
Hypothesis
Ref Expression
abeq1i.1 {𝑥𝜑} = 𝐴
Assertion
Ref Expression
abeq1i (𝜑𝑥𝐴)

Proof of Theorem abeq1i
StepHypRef Expression
1 abeq1i.1 . . . 4 {𝑥𝜑} = 𝐴
21eqcomi 2769 . . 3 𝐴 = {𝑥𝜑}
32abeq2i 2873 . 2 (𝑥𝐴𝜑)
43bicomi 214 1 (𝜑𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1632  wcel 2139  {cab 2746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1635  df-ex 1854  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756
This theorem is referenced by: (None)
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