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Theorem sbequ12r 1783
Description: An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
Assertion
Ref Expression
sbequ12r (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))

Proof of Theorem sbequ12r
StepHypRef Expression
1 sbequ12 1782 . . 3 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
21bicomd 141 . 2 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
32equcoms 1719 1 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541
This theorem depends on definitions:  df-bi 117  df-sb 1774
This theorem is referenced by:  abbi  2307  findes  4635  opeliunxp  4714  isarep1  5340  bezoutlemmain  12135
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