Theorem sbn 2284

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 2065. (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.

Step	Hyp	Ref	Expression
1		df-sb 2065	. 2 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
2		alinexa 1841	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
3	2	imbi2i 336	. . 3 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ (𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
4	3	albii 1817	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
5		alinexa 1841	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ¬ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
6		dfsb7 2283	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
7	5, 6	xchbinxr 335	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ¬ [𝑡 / 𝑥]𝜑)
8	1, 4, 7	3bitri 297	1 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by:  sbex  2285  sbi2  2306  sbor  2311  sbcng  3855  difab  4329  difopab  5854  wl-sb8eft  37505  wl-sb8et  37507  pm13.196a  44383  ichn  47330