Theorem bj-denoteslem 35055

Description: Lemma for bj-denotes 35056. (Contributed by BJ, 24-Apr-2024.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-denoteslem	⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem bj-denoteslem

Step	Hyp	Ref	Expression
1		vextru 2722	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ ⊤}
2	1	biantru 530	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
3	2	exbii 1850	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
4		dfclel 2817	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ⊤} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
5	3, 4	bitr4i 277	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ⊤wtru 1540  ∃wex 1782   ∈ wcel 2106  {cab 2715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-clel 2816

This theorem is referenced by:  bj-denotes  35056  bj-issettru  35057  bj-elabtru  35058