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Theorem bj-sbfvv 33129
Description: Version of sbf 2471 with two disjoint variable conditions, which does not require ax-10 2183 nor ax-13 2352. (Contributed by BJ, 1-May-2021.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sbfvv ([𝑦 / 𝑥]𝜑𝜑)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem bj-sbfvv
StepHypRef Expression
1 spsbe 2066 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2 19.9v 2078 . . 3 (∃𝑥𝜑𝜑)
31, 2sylib 209 . 2 ([𝑦 / 𝑥]𝜑𝜑)
4 ax-5 2005 . . 3 (𝜑 → ∀𝑥𝜑)
5 bj-stdpc4v 33119 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
64, 5syl 17 . 2 (𝜑 → [𝑦 / 𝑥]𝜑)
73, 6impbii 200 1 ([𝑦 / 𝑥]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wal 1650  wex 1874  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-sb 2063
This theorem is referenced by:  bj-vjust2  33374
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