Theorem bj-denoteslem 36850

Description: Duplicate of issettru 2818 and bj-issettruALTV 36852.

Lemma for bj-denotesALTV 36851. (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 2720	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ ⊤}
2	1	biantru 529	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
3	2	exbii 1848	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
4		dfclel 2816	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ⊤} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
5	3, 4	bitr4i 278	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541  ∃wex 1779   ∈ wcel 2108  {cab 2713

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-clel 2815

This theorem is referenced by:  bj-denotesALTV  36851