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Theorem sbequ12r 2109
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 2108 . . 3 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
21bicomd 213 . 2 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
32equcoms 1944 1 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1878
This theorem is referenced by:  sbequ12a  2110  sbid  2111  sb5rf  2421  sb6rf  2422  2sb5rf  2450  2sb6rf  2451  opeliunxp  5141  isarep1  5945  findes  7058  axrepndlem1  9374  axrepndlem2  9375  nn0min  29450  esumcvg  29971  bj-abbi  32471  bj-sbidmOLD  32529  wl-nfs1t  32995  wl-sb6rft  33001  wl-equsb4  33009  wl-ax11-lem5  33037  sbcalf  33588  sbcexf  33589  opeliun2xp  41429
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