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Theorem sb8euv 2663
 Description: Variable substitution in unique existential quantifier. Version of sb8eu 2664 requiring more disjoint variables, but fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Wolf Lammen, 7-Feb-2023.)
Hypothesis
Ref Expression
sb8euv.nf 𝑦𝜑
Assertion
Ref Expression
sb8euv (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb8euv
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sb8euv.nf . . 3 𝑦𝜑
21nfsbv 2341 . 2 𝑦[𝑤 / 𝑥]𝜑
32sb8eulem 2662 1 (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  Ⅎwnf 1785  [wsb 2069  ∃!weu 2631 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-10 2143  ax-11 2159  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632 This theorem is referenced by:  cbvmowOLD  2667  cbveuwOLD  2670  eu1  2674  cbvreuw  3392
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