Theorem issetft 3487

Description: Closed theorem form of isset 3486 that does not require 𝑥 and 𝐴 to be distinct. Extracted from the proof of vtoclgft 3540. (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 3486	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2		nfv 1916	. . . . 5 ⊢ Ⅎ𝑦Ⅎ𝑥𝐴
3		nfnfc1 2905	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
4		nfcvd 2903	. . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦)
5		id 22	. . . . . . 7 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
6	4, 5	nfeqd 2912	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
7	6	nfnd 1860	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴)
8		nfvd 1917	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴)
9		eqeq1 2735	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝐴 ↔ 𝑥 = 𝐴))
10	9	notbid 318	. . . . . 6 ⊢ (𝑦 = 𝑥 → (¬ 𝑦 = 𝐴 ↔ ¬ 𝑥 = 𝐴))
11	10	a1i 11	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → (𝑦 = 𝑥 → (¬ 𝑦 = 𝐴 ↔ ¬ 𝑥 = 𝐴)))
12	2, 3, 7, 8, 11	cbv2w 2332	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ∀𝑥 ¬ 𝑥 = 𝐴))
13		alnex 1782	. . . 4 ⊢ (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴)
14		alnex 1782	. . . 4 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
15	12, 13, 14	3bitr3g 313	. . 3 ⊢ (Ⅎ𝑥𝐴 → (¬ ∃𝑦 𝑦 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴))
16	15	con4bid 317	. 2 ⊢ (Ⅎ𝑥𝐴 → (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴))
17	1, 16	bitrid 283	1 ⊢ (Ⅎ𝑥𝐴 → (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105  Ⅎwnfc 2882  Vcvv 3473

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-v 3475

This theorem is referenced by:  issetf  3488  vtoclgft  3540