Theorem bj-denoteslem 34792

Description: Lemma for bj-denotes 34793. (Contributed by BJ, 24-Apr-2024.)

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 2721	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ ⊤}
2	1	biantru 533	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
3	2	exbii 1855	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
4		dfclel 2817	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ⊤} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
5	3, 4	bitr4i 281	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543  ⊤wtru 1544  ∃wex 1787   ∈ wcel 2110  {cab 2714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-clel 2816

This theorem is referenced by:  bj-denotes  34793  bj-issettru  34794  bj-elabtru  34795