Theorem issetft 3452

Description: Closed theorem form of isset 3450 that does not require 𝑥 and 𝐴 to be distinct. Extracted from the proof of vtoclgft 3505. (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 3450	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2		cbvexeqsetf 3451	. 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 1541  ∃wex 1780   ∈ wcel 2111  Ⅎwnfc 2879  Vcvv 3436

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-v 3438

This theorem is referenced by:  issetf  3453