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Theorem abeqabi 41754
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 abbi 2809 . 2 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
42, 3bitr4i 278 1 ({𝑥𝜑} = 𝐴 ↔ ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1540   = wceq 1542  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729
This theorem is referenced by:  abpr  41755  abtp  41756
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