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Theorem sb8eu 2685
 Description: Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2390. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2684. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.) (New usage is discouraged.)
Hypothesis
Ref Expression
sb8eu.1 𝑦𝜑
Assertion
Ref Expression
sb8eu (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8eu
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sb8eu.1 . . 3 𝑦𝜑
21nfsb 2565 . 2 𝑦[𝑤 / 𝑥]𝜑
32sb8eulem 2683 1 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208  Ⅎwnf 1784  [wsb 2069  ∃!weu 2652 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653 This theorem is referenced by:  sb8mo  2686  cbveu  2690  cbvreu  3426
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