Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ssbft Structured version   Visualization version   GIF version

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  wnf 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)
  Copyright terms: Public domain W3C validator