Theorem abeq2f 2949

Description: Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a non-free variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) Avoid ax-13 2379. (Revised by Wolf Lammen, 13-May-2023.)

Hypothesis

Ref	Expression
abeq2f.0	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
abeq2f	⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))

Proof of Theorem abeq2f

Step	Hyp	Ref	Expression
1		abeq2f.0	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfab1 2921	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
3	1, 2	cleqf 2947	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
4		abid 2739	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
5	4	bibi2i 341	. . 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 278	1 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))

Syntax hints:   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735  Ⅎwnfc 2899

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901

This theorem is referenced by:  rabid2f  3300  mptfnf  6466