Theorem issetft 3494

Description: Closed theorem form of isset 3492 that does not require 𝑥 and 𝐴 to be distinct. Extracted from the proof of vtoclgft 3552. (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 3492	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2		cbvexeqsetf 3493	. 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 1776   ∈ wcel 2106  Ⅎwnfc 2888  Vcvv 3478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-v 3480

This theorem is referenced by:  issetf  3495