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Theorem sbv 2095
 Description: Substitution for a variable not occurring in a proposition. See sbf 2268 for a version without disjoint variable condition on 𝑥, 𝜑. If one adds a disjoint variable condition on 𝑥, 𝑡, then sbv 2095 can be proved directly by chaining equsv 2009 with sb6 2090. (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
sbv ([𝑡 / 𝑥]𝜑𝜑)
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑡)

Proof of Theorem sbv
StepHypRef Expression
1 spsbe 2087 . . 3 ([𝑡 / 𝑥]𝜑 → ∃𝑥𝜑)
2 ax5e 1913 . . 3 (∃𝑥𝜑𝜑)
31, 2syl 17 . 2 ([𝑡 / 𝑥]𝜑𝜑)
4 ax-5 1911 . . 3 (𝜑 → ∀𝑥𝜑)
5 stdpc4 2073 . . 3 (∀𝑥𝜑 → [𝑡 / 𝑥]𝜑)
64, 5syl 17 . 2 (𝜑 → [𝑡 / 𝑥]𝜑)
73, 6impbii 212 1 ([𝑡 / 𝑥]𝜑𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wal 1536  ∃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 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-sb 2070 This theorem is referenced by:  sbcom4  2096  sbievw2  2104  csb0  4314  iuninc  30334  measiuns  31601  ballotlemodife  31880  bj-vjust  34489  mptsnunlem  34774  ichv  44009  ichf  44010
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