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Theorem bj-ssbft 32767
 Description: See sbft 2407. This proof is from Tarski's FOL together with sp 2091 (and its dual). (Contributed by BJ, 22-Dec-2020.)
Assertion
Ref Expression
bj-ssbft (Ⅎ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))

Proof of Theorem bj-ssbft
StepHypRef Expression
1 bj-sbex 32751 . . 3 ([𝑡/𝑥]b𝜑 → ∃𝑥𝜑)
2 df-nf 1750 . . . 4 (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
32biimpi 206 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑))
4 sp 2091 . . 3 (∀𝑥𝜑𝜑)
51, 3, 4syl56 36 . 2 (Ⅎ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))
6 19.8a 2090 . . 3 (𝜑 → ∃𝑥𝜑)
7 bj-alsb 32750 . . 3 (∀𝑥𝜑 → [𝑡/𝑥]b𝜑)
86, 3, 7syl56 36 . 2 (Ⅎ𝑥𝜑 → (𝜑 → [𝑡/𝑥]b𝜑))
95, 8impbid 202 1 (Ⅎ𝑥𝜑 → ([𝑡/𝑥]b𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521  ∃wex 1744  Ⅎwnf 1748  [wssb 32744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087 This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750  df-ssb 32745 This theorem is referenced by: (None)
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