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Theorem abeqabi 43421
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 2750 . 2 ({𝑥𝜑} = 𝐴 ↔ {𝑥𝜑} = {𝑥𝜓})
3 abbib 2811 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
42, 3bitri 275 1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1538   = wceq 1540  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729
This theorem is referenced by:  abpr  43422  abtp  43423
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