Theorem issetft 3504

Description: Closed theorem form of isset 3502 that does not require 𝑥 and 𝐴 to be distinct. Extracted from the proof of vtoclgft 3564. (Contributed by Wolf Lammen, 9-Apr-2025.)

Assertion

Ref	Expression
issetft	⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))

Proof of Theorem issetft

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isset 3502	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2		cbvexeqsetf 3503	. 2 ⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
3	1, 2	bitr4id 290	1 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Ⅎwnfc 2893  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-v 3490

This theorem is referenced by:  issetf  3505