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Theorem equequ1 1684
 Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
Assertion
Ref Expression
equequ1 (x = y → (x = zy = z))

Proof of Theorem equequ1
StepHypRef Expression
1 ax-8 1675 . 2 (x = y → (x = zy = z))
2 equtr 1682 . 2 (x = y → (y = zx = z))
31, 2impbid 183 1 (x = y → (x = zy = z))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176 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 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by:  equequ2  1686  ax12olem6  1932  ax10lem2  1937  ax10lem4  1941  equveli  1988  dveeq1  2018  drsb1  2022  equsb3lem  2101  dveeq1-o  2187  dveeq1-o16  2188  ax10-16  2190  ax11eq  2193  2mo  2282  2eu6  2289  euequ1  2292  axext3  2336  cbviota  4344
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