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Theorem sbequ12r 1671
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 1670 . . 3 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
21bicomd 133 . 2 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
32equcoms 1610 1 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  abbi  2167  findes  4354  opeliunxp  4423  isarep1  5013
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