MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbn Structured version   Visualization version   GIF version

Theorem sbn 2280
Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) Revise df-sb 2071. (Revised by BJ, 25-Dec-2020.)
Assertion
Ref Expression
sbn ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑)

Proof of Theorem sbn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-sb 2071 . 2 ([𝑡 / 𝑥] ¬ 𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
2 alinexa 1848 . . . 4 (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦𝜑))
32imbi2i 335 . . 3 ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ (𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)))
43albii 1825 . 2 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)))
5 alinexa 1848 . . 3 (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ¬ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
6 dfsb7 2279 . . 3 ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
75, 6xchbinxr 334 . 2 (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ¬ [𝑡 / 𝑥]𝜑)
81, 4, 73bitri 296 1 ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wal 1539  wex 1785  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-10 2140  ax-12 2174
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1786  df-nf 1790  df-sb 2071
This theorem is referenced by:  sbex  2281  sbi2  2302  sbor  2307  sbcng  3769  difab  4239  wl-sb8et  35687  pm13.196a  41985  ichn  44860
  Copyright terms: Public domain W3C validator