Theorem eqvincf 1408

Description: A variable introduction law for class equality, requiring only that x not be free in A and B (instead of not occurring in them).

Hypotheses

Ref	Expression
eqvincf.1	⊢ (y ∈ A → ∀x y ∈ A)
eqvincf.2	⊢ (y ∈ B → ∀x y ∈ B)
eqvincf.3	⊢ A ∈ V

Assertion

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

Distinct variable group(s):   y,A   y,B   x,y

Proof of Theorem eqvincf

Step	Hyp	Ref	Expression
1		eqvincf.3	. . 3 ⊢ A ∈ V
2	1	eqvinc 1407	. 2 ⊢ (A = B ↔ ∃y(y = A ∧ y = B))
3		eqvincf.1	. . . . 5 ⊢ (y ∈ A → ∀x y ∈ A)
4	3	hbeleq 1173	. . . 4 ⊢ (y = A → ∀x y = A)
5		eqvincf.2	. . . . 5 ⊢ (y ∈ B → ∀x y ∈ B)
6	5	hbeleq 1173	. . . 4 ⊢ (y = B → ∀x y = B)
7	4, 6	hban 704	. . 3 ⊢ ((y = A ∧ y = B) → ∀x(y = A ∧ y = B))
8		ax-17 925	. . 3 ⊢ ((x = A ∧ x = B) → ∀y(x = A ∧ x = B))
9		cleq1 1107	. . . 4 ⊢ (y = x → (y = A ↔ x = A))
10		cleq1 1107	. . . 4 ⊢ (y = x → (y = B ↔ x = B))
11	9, 10	anbi12d 476	. . 3 ⊢ (y = x → ((y = A ∧ y = B) ↔ (x = A ∧ x = B)))
12	7, 8, 11	cbvex 849	. 2 ⊢ (∃y(y = A ∧ y = B) ↔ ∃x(x = A ∧ x = B))
13	2, 12	bitr 151	1 ⊢ (A = B ↔ ∃x(x = A ∧ x = B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  fvopabgf 2874  fvopabnf 2875

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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