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Theorem sbequ2 2066
Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)
Assertion
Ref Expression
sbequ2 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))

Proof of Theorem sbequ2
StepHypRef Expression
1 df-sb 2065 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21simplbi 492 . 2 ([𝑦 / 𝑥]𝜑 → (𝑥 = 𝑦𝜑))
32com12 32 1 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wex 1875  [wsb 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-an 386  df-sb 2065
This theorem is referenced by:  stdpc7  2128  sbequ12  2278  dfsb2  2490  sbequi  2492  sbi1  2509  bj-mo3OLD  33327  2pm13.193  39538  2pm13.193VD  39899
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