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Theorem equvin 1740
 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 1638 . 2 (x = yz(x = z z = y))
2 ax-17 1416 . . 3 (x = yz x = y)
3 equtr 1592 . . . 4 (x = z → (z = yx = y))
43imp 115 . . 3 ((x = z z = y) → x = y)
52, 4exlimih 1481 . 2 (z(x = z z = y) → x = y)
61, 5impbii 117 1 (x = yz(x = z z = y))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424 This theorem depends on definitions:  df-bi 110 This theorem is referenced by: (None)
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