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Theorem sbequ2 1779
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbequ2 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))

Proof of Theorem sbequ2
StepHypRef Expression
1 df-sb 1773 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
2 simpl 109 . . 3 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → (𝑥 = 𝑦𝜑))
32com12 30 . 2 (𝑥 = 𝑦 → (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → 𝜑))
41, 3biimtrid 152 1 (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wex 1502  [wsb 1772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106
This theorem depends on definitions:  df-bi 117  df-sb 1773
This theorem is referenced by:  stdpc7  1780  sbequ12  1781  sbequi  1849  mo23  2077  mopick  2114
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