Theorem issetlem 2814

Description: Lemma for elisset 2816 and isset 3488. (Contributed by NM, 26-May-1993.) Extract from the proof of isset 3488. (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 2812	. 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
2		issetlem.1	. . . 4 ⊢ 𝑥 ∈ 𝑉
3	2	biantru 531	. . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
4	3	exbii 1851	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉))
5	1, 4	bitr4i 278	1 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥 𝑥 = 𝐴)

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-clel 2811

This theorem is referenced by:  elisset  2816  isset  3488