Theorem eqvinc 2967

Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
eqvinc.1	⊢ A ∈ V

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem eqvinc

Step	Hyp	Ref	Expression
1		eqvinc.1	. . . . 5 ⊢ A ∈ V
2	1	isseti 2866	. . . 4 ⊢ ∃x x = A
3		ax-1 6	. . . . . 6 ⊢ (x = A → (A = B → x = A))
4		eqtr 2370	. . . . . . 7 ⊢ ((x = A ∧ A = B) → x = B)
5	4	ex 423	. . . . . 6 ⊢ (x = A → (A = B → x = B))
6	3, 5	jca 518	. . . . 5 ⊢ (x = A → ((A = B → x = A) ∧ (A = B → x = B)))
7	6	eximi 1576	. . . 4 ⊢ (∃x x = A → ∃x((A = B → x = A) ∧ (A = B → x = B)))
8		pm3.43 832	. . . . 5 ⊢ (((A = B → x = A) ∧ (A = B → x = B)) → (A = B → (x = A ∧ x = B)))
9	8	eximi 1576	. . . 4 ⊢ (∃x((A = B → x = A) ∧ (A = B → x = B)) → ∃x(A = B → (x = A ∧ x = B)))
10	2, 7, 9	mp2b 9	. . 3 ⊢ ∃x(A = B → (x = A ∧ x = B))
11	10	19.37aiv 1900	. 2 ⊢ (A = B → ∃x(x = A ∧ x = B))
12		eqtr2 2371	. . 3 ⊢ ((x = A ∧ x = B) → A = B)
13	12	exlimiv 1634	. 2 ⊢ (∃x(x = A ∧ x = B) → A = B)
14	11, 13	impbii 180	1 ⊢ (A = B ↔ ∃x(x = A ∧ x = B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  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-11 1746  ax-ext 2334

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

This theorem is referenced by:  eqvincf  2968  preaddccan2lem1  4455  dff13  5472  nncdiv3lem1  6276