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Theorem abeqabi 42716
Description: Generalized condition for a class abstraction to be equal to some class. (Contributed by RP, 2-Sep-2024.)
Hypothesis
Ref Expression
abeqabi.a 𝐴 = {𝑥𝜓}
Assertion
Ref Expression
abeqabi ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝜓))

Proof of Theorem abeqabi
StepHypRef Expression
1 abeqabi.a . . 3 𝐴 = {𝑥𝜓}
21eqeq2i 2739 . 2 ({𝑥𝜑} = 𝐴 ↔ {𝑥𝜑} = {𝑥𝜓})
3 abbib 2798 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
42, 3bitri 275 1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1531   = wceq 1533  {cab 2703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718
This theorem is referenced by:  abpr  42717  abtp  42718
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