Theorem abeqabi 44025

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 = {𝑥 ∣ 𝜓}

Assertion

Ref	Expression
abeqabi	⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝜓))

Proof of Theorem abeqabi

Step	Hyp	Ref	Expression
1		abeqabi.a	. . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜓}
2	1	eqeq2i 2782	. 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
3		abbib 2838	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
4	2, 3	bitri 278	1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝜓))

Syntax hints:   ↔ wb 209  ∀wal 1565   = wceq 1567  {cab 2747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761

This theorem is referenced by:  abpr  44026  abtp  44027