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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 = yz(x = z z = y))
Distinct variable groups:   x,z   y,z

Proof of Theorem equvin
StepHypRef Expression
1 equvini 1987 . 2 (x = yz(x = z z = y))
2 equtr 1682 . . . 4 (x = z → (z = yx = y))
32imp 418 . . 3 ((x = z z = y) → x = y)
43exlimiv 1634 . 2 (z(x = z z = y) → x = y)
51, 4impbii 180 1 (x = yz(x = z z = y))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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)
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