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Theorem sbequ12a 2115
Description: An equality theorem for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Jun-2019.)
Assertion
Ref Expression
sbequ12a (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))

Proof of Theorem sbequ12a
StepHypRef Expression
1 sbequ12r 2114 . 2 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
2 sbequ12 2113 . 2 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
31, 2bitr2d 269 1 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  [wsb 1882
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 1841  ax-6 1890  ax-7 1937  ax-12 2049
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1883
This theorem is referenced by:  sbco3  2421  sb9  2430
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