Theorem cbvexeqsetf 3473

Description: The expression ∃𝑥𝑥 = 𝐴 means "𝐴 is a set" even if 𝐴 contains 𝑥 as a bound variable. This lemma helps minimizing axiom or df-clab 2745 usage in some cases. Extracted from the proof of issetft 3474. (Contributed by Wolf Lammen, 30-Jul-2025.)

Assertion

Ref	Expression
cbvexeqsetf	⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem cbvexeqsetf

Step	Hyp	Ref	Expression
1		nfnfc1 2931	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
2		nfv 1947	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑥𝐴
3		nfvd 1948	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑦 ¬ 𝑥 = 𝐴)
4		nfcvd 2929	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝑦)
5		id 23	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥𝐴)
6	4, 5	nfeqd 2938	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
7	6	nfnd 1891	. . . 4 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 ¬ 𝑦 = 𝐴)
8		eqeq1 2770	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
9	8	notbid 321	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 = 𝐴 ↔ ¬ 𝑦 = 𝐴))
10	9	a1i 11	. . . 4 ⊢ (Ⅎ𝑥𝐴 → (𝑥 = 𝑦 → (¬ 𝑥 = 𝐴 ↔ ¬ 𝑦 = 𝐴)))
11	1, 2, 3, 7, 10	cbv2w 2372	. . 3 ⊢ (Ⅎ𝑥𝐴 → (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑦 ¬ 𝑦 = 𝐴))
12		alnex 1814	. . 3 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
13		alnex 1814	. . 3 ⊢ (∀𝑦 ¬ 𝑦 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴)
14	11, 12, 13	3bitr3g 316	. 2 ⊢ (Ⅎ𝑥𝐴 → (¬ ∃𝑥 𝑥 = 𝐴 ↔ ¬ ∃𝑦 𝑦 = 𝐴))
15	14	con4bid 320	1 ⊢ (Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1812  Ⅎwnfc 2913

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-cleq 2758  df-nfc 2915

This theorem is used by:  issetft  3474  spcimgft  3518