Theorem bj-clex 37032

Description: Two ways of stating that a class is a set. (Contributed by BJ, 18-Jan-2025.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-clex.1	⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)

Assertion

Ref	Expression
bj-clex	⊢ (𝐴 ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-clex

Step	Hyp	Ref	Expression
1		isset 3494	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2		dfcleq 2730	. . . 4 ⊢ (𝑦 = 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴))
3		bj-clex.1	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
4	3	bibi2i 337	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝑦 ↔ 𝜑))
5	4	albii 1819	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
6	2, 5	bitri 275	. . 3 ⊢ (𝑦 = 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
7	6	exbii 1848	. 2 ⊢ (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))
8	1, 7	bitri 275	1 ⊢ (𝐴 ∈ V ↔ ∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑))

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482

This theorem is referenced by:  bj-axsn  37033  bj-axbun  37037  bj-axadj  37042