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Theorem bj-ssbft 31831
 Description: See sbft 2367. This proof is from Tarski's FOL together with sp 2041 (and its dual). (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssbft (ℲℲ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))

Proof of Theorem bj-ssbft
StepHypRef Expression
1 bj-sbex 31815 . . 3 ([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)
2 df-bj-nf 31765 . . . 4 (ℲℲ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
32biimpi 205 . . 3 (ℲℲ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4 sp 2041 . . 3 (∀𝑥𝜑𝜑)
51, 3, 4syl56 35 . 2 (ℲℲ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))
6 19.8a 2039 . . 3 (𝜑 → ∃𝑥𝜑)
7 bj-alsb 31814 . . 3 (∀𝑥𝜑 → [𝑡/𝑥]b𝜑)
86, 3, 7syl56 35 . 2 (ℲℲ𝑥𝜑 → (𝜑 → [𝑡/𝑥]b𝜑))
95, 8impbid 201 1 (ℲℲ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  ℲℲwnff 31764  [wssb 31808 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-ex 1696  df-bj-nf 31765  df-ssb 31809 This theorem is referenced by: (None)
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