Theorem bj-denoteslem 34201

Description: Lemma for bj-denotes 34202. (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 2806	. . . 4 ⊢ 𝑥 ∈ {𝑦 ∣ ⊤}
2	1	biantru 532	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
3	2	exbii 1848	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
4		dfclel 2894	. 2 ⊢ (𝐴 ∈ {𝑦 ∣ ⊤} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ {𝑦 ∣ ⊤}))
5	3, 4	bitr4i 280	1 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537  ⊤wtru 1538  ∃wex 1780   ∈ wcel 2114  {cab 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2800  df-clel 2893

This theorem is referenced by:  bj-denotes  34202  bj-issettru  34203  bj-elabtru  34204