Theorem eqvincf 2968

Description: A variable introduction law for class equality, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Sep-2003.)

Hypotheses

Ref	Expression
eqvincf.1	⊢ ℲxA
eqvincf.2	⊢ ℲxB
eqvincf.3	⊢ A ∈ V

Assertion

Ref	Expression
eqvincf	⊢ (A = B ↔ ∃x(x = A ∧ x = B))

Proof of Theorem eqvincf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqvincf.3	. . 3 ⊢ A ∈ V
2	1	eqvinc 2967	. 2 ⊢ (A = B ↔ ∃y(y = A ∧ y = B))
3		eqvincf.1	. . . . 5 ⊢ ℲxA
4	3	nfeq2 2501	. . . 4 ⊢ Ⅎx y = A
5		eqvincf.2	. . . . 5 ⊢ ℲxB
6	5	nfeq2 2501	. . . 4 ⊢ Ⅎx y = B
7	4, 6	nfan 1824	. . 3 ⊢ Ⅎx(y = A ∧ y = B)
8		nfv 1619	. . 3 ⊢ Ⅎy(x = A ∧ x = B)
9		eqeq1 2359	. . . 4 ⊢ (y = x → (y = A ↔ x = A))
10		eqeq1 2359	. . . 4 ⊢ (y = x → (y = B ↔ x = B))
11	9, 10	anbi12d 691	. . 3 ⊢ (y = x → ((y = A ∧ y = B) ↔ (x = A ∧ x = B)))
12	7, 8, 11	cbvex 1985	. 2 ⊢ (∃y(y = A ∧ y = B) ↔ ∃x(x = A ∧ x = B))
13	2, 12	bitri 240	1 ⊢ (A = B ↔ ∃x(x = A ∧ x = B))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by: (None)