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Theorem equequ2 1640
 Description: An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
equequ2 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Proof of Theorem equequ2
StepHypRef Expression
1 equtrr 1637 . 2 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
2 equtrr 1637 . . 3 (𝑦 = 𝑥 → (𝑧 = 𝑦𝑧 = 𝑥))
32equcoms 1635 . 2 (𝑥 = 𝑦 → (𝑧 = 𝑦𝑧 = 𝑥))
41, 3impbid 127 1 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1379  ax-ie2 1424  ax-8 1436  ax-17 1460  ax-i9 1464 This theorem depends on definitions:  df-bi 115 This theorem is referenced by:  ax11v2  1742  ax11v  1749  ax11ev  1750  equs5or  1752  eujust  1944  euf  1947  mo23  1983  iotaval  4908  dffun4f  4948  dff13f  5441  supmoti  6465  isoti  6479
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