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Theorem bj-sbfvv 31953
Description: Version of sbf 2368 with two dv conditions, which does not require ax-10 2006 nor ax-13 2234. (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 1871 . . 3 ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2 19.9v 1883 . . 3 (∃𝑥𝜑𝜑)
31, 2sylib 207 . 2 ([𝑦 / 𝑥]𝜑𝜑)
4 ax-5 1827 . . 3 (𝜑 → ∀𝑥𝜑)
5 bj-stdpc4v 31942 . . 3 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
64, 5syl 17 . 2 (𝜑 → [𝑦 / 𝑥]𝜑)
73, 6impbii 198 1 ([𝑦 / 𝑥]𝜑𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 195  wal 1473  wex 1695  [wsb 1867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868
This theorem is referenced by:  bj-vjust2  32206
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