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 = {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 2750	. 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
3		abbib 2811	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
4	2, 3	bitri 275	1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝜓))

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