Theorem abeqabi 43984

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 2775	. 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
3		abbib 2831	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
4	2, 3	bitri 277	1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝜓))

Syntax hints:   ↔ wb 208  ∀wal 1558   = wceq 1560  {cab 2740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754

This theorem is referenced by:  abpr  43985  abtp  43986