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Theorem sbequ12r 2150
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 2149 . . 3 (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
21bicomd 213 . 2 (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑𝜑))
32equcoms 1993 1 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-sb 1938
This theorem is referenced by:  sbequ12a  2151  sbid  2152  sb5rf  2450  sb6rf  2451  2sb5rf  2479  2sb6rf  2480  opeliunxp  5204  isarep1  6015  findes  7138  axrepndlem1  9452  axrepndlem2  9453  nn0min  29695  esumcvg  30276  bj-abbi  32900  bj-sbidmOLD  32956  wl-nfs1t  33454  wl-sb6rft  33460  wl-equsb4  33468  wl-ax11-lem5  33496  sbcalf  34047  sbcexf  34048  opeliun2xp  42436
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