Theorem eqabf 2927

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 2375. (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 2899	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3	1, 2	cleqf 2926	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
4		abid 2716	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
5	4	bibi2i 337	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑))
6	5	albii 1818	. 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 1537   = wceq 1539   ∈ wcel 2107  {cab 2712  Ⅎwnfc 2882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884

This theorem is referenced by:  abid2f  2928  rabid2f  3452  mptfnf  6684