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Theorem equequ2 1646
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 1643 . 2 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
2 equtrr 1643 . . 3 (𝑦 = 𝑥 → (𝑧 = 𝑦𝑧 = 𝑥))
32equcoms 1641 . 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 1383  ax-ie2 1428  ax-8 1440  ax-17 1464  ax-i9 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  ax11v2  1748  ax11v  1755  ax11ev  1756  equs5or  1758  eujust  1950  euf  1953  mo23  1989  eleq1w  2148  disjiun  3840  iotaval  4991  dffun4f  5031  dff13f  5549  supmoti  6686  isoti  6700
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