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Theorem sbelx 2252
 Description: Elimination of substitution. Also see sbel2x 2487. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2379. (Revised by Wolf Lammen, 6-Aug-2023.) Avoid ax-10 2142. (Revised by Gino Giotto, 20-Aug-2023.)
Assertion
Ref Expression
sbelx (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem sbelx
StepHypRef Expression
1 sbequ12r 2251 . . 3 (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))
21equsexvw 2011 . 2 (∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑) ↔ 𝜑)
32bicomi 227 1 (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1781  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070 This theorem is referenced by:  pm13.196a  41491
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