Theorem bj-clex 37207

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 3455	. 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 1821	. . . 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 1850	. 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 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3441

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-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443

This theorem is referenced by:  bj-axsn  37208  bj-axbun  37212  bj-axadj  37217