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Theorem sb8eu 2602
Description: Variable substitution in unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2374. For a version requiring more disjoint variables, but fewer axioms, see sb8euv 2601. (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 2529 . 2 𝑦[𝑤 / 𝑥]𝜑
32sb8eulem 2600 1 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wnf 1790  [wsb 2071  ∃!weu 2570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2374
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571
This theorem is referenced by:  sb8mo  2603  cbveu  2611  cbvreu  3379
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