Theorem abeqabi 44549

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 2776	. 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
3		abbib 2832	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
4	2, 3	bitri 278	1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝜓))

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567   = wceq 1569  {cab 2741

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755

This theorem is used by:  abpr  44550  abtp  44551