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Theorem sbn 2287
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 2070. (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 2070 . 2 ([𝑡 / 𝑥] ¬ 𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
2 alinexa 1843 . . . 4 (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦𝜑))
32imbi2i 338 . . 3 ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ (𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)))
43albii 1820 . 2 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)))
5 alinexa 1843 . . 3 (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ¬ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
6 dfsb7 2285 . . 3 ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
75, 6xchbinxr 337 . 2 (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦𝜑)) ↔ ¬ [𝑡 / 𝑥]𝜑)
81, 4, 73bitri 299 1 ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1535  wex 1780  [wsb 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070
This theorem is referenced by:  sbex  2288  sbi2  2310  sbanOLD  2312  sbor  2316  sbanvOLD  2326  sbcng  3798  difab  4250  bj-ab0  34241  wl-sb8et  34830  wl-dfrexsb  34892  pm13.196a  40901  ichn  43764
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