Theorem eqabf 2929

Description: Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a nonfree variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) Avoid ax-13 2377. (Revised by Wolf Lammen, 13-May-2023.)

Hypothesis

Ref	Expression
eqabf.0	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
eqabf	⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))

Proof of Theorem eqabf

Step	Hyp	Ref	Expression
1		eqabf.0	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfab1 2901	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3	1, 2	cleqf 2928	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
4		abid 2719	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
5	4	bibi2i 337	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑))
6	5	albii 1821	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
7	3, 6	bitri 275	1 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))

Syntax hints:   ↔ wb 206  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886

This theorem is referenced by:  abid2f  2930  rabid2f  3432  mptfnf  6635