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Theorem equequ2 1686
 Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.)
Assertion
Ref Expression
equequ2 (x = y → (z = xz = y))

Proof of Theorem equequ2
StepHypRef Expression
1 equequ1 1684 . 2 (x = y → (x = zy = z))
2 equcom 1680 . 2 (x = zz = x)
3 equcom 1680 . 2 (y = zz = y)
41, 2, 33bitr3g 278 1 (x = y → (z = xz = y))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176 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 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by:  ax10lem2  1937  dveeq2  1940  ax10lem4  1941  ax9  1949  ax11v2  1992  ax11vALT  2097  dveeq2-o  2184  dveeq2-o16  2185  ax10-16  2190  ax11eq  2193  ax11inda  2200  ax11v2-o  2201  eujust  2206  eujustALT  2207  euf  2210  mo  2226  2mo  2282  2eu6  2289  iotaval  4350  nndisjeq  4429  dfid3  4768
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