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Theorem abeq1i 2739
 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 2635 . . 3 𝐴 = {𝑥𝜑}
32abeq2i 2738 . 2 (𝑥𝐴𝜑)
43bicomi 214 1 (𝜑𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1480   ∈ wcel 1992  {cab 2612 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-12 2049  ax-ext 2606 This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1483  df-ex 1702  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622 This theorem is referenced by: (None)
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