Theorem bj-denoteslem 36240

Description: Lemma for bj-denotes 36241. (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 2708	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ ⊤}
2	1	biantru 529	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
3	2	exbii 1842	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
4		dfclel 2803	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ⊤} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
5	3, 4	bitr4i 278	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533  ⊤wtru 1534  ∃wex 1773   ∈ wcel 2098  {cab 2701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-clel 2802

This theorem is referenced by:  bj-denotes  36241  bj-issettru  36242  bj-elabtru  36243