Theorem issetlem 2845

Description: Lemma for elisset 2847 and isset 3471. (Contributed by NM, 26-May-1993.) Extract from the proof of isset 3471. (Revised by WL, 2-Feb-2025.)

Hypothesis

Ref	Expression
issetlem.1	⊢ 𝑥 ∈ 𝑉

Assertion

Ref	Expression
issetlem	⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥 𝑥 = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Proof of Theorem issetlem

Step	Hyp	Ref	Expression
1		dfclel 2841	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		issetlem.1	. . . 4 ⊢ 𝑥 ∈ 𝑉
3	2	biantru 539	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
4	3	exbii 1881	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
5	1, 4	bitr4i 281	1 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥 𝑥 = 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2840

This theorem is used by:  isset  3471