Theorem equvin 2001

Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
equvin	⊢ (x = y ↔ ∃z(x = z ∧ z = y))

Distinct variable groups:   x,z   y,z

Proof of Theorem equvin

Step	Hyp	Ref	Expression
1		equvini 1987	. 2 ⊢ (x = y → ∃z(x = z ∧ z = y))
2		equtr 1682	. . . 4 ⊢ (x = z → (z = y → x = y))
3	2	imp 418	. . 3 ⊢ ((x = z ∧ z = y) → x = y)
4	3	exlimiv 1634	. 2 ⊢ (∃z(x = z ∧ z = y) → x = y)
5	1, 4	impbii 180	1 ⊢ (x = y ↔ ∃z(x = z ∧ z = y))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541

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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)