Theorem sbn 2279

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 2063. (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 2063	. 2 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)))
2		alinexa 1840	. . . 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 1816	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
5		alinexa 1840	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ¬ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
6		dfsb7 2278	. . 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 1776  [wsb 2062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-sb 2063

This theorem is referenced by:  sbex  2280  sbi2  2301  sbor  2306  sbcng  3842  difab  4316  difopab  5843  wl-sb8eft  37532  wl-sb8et  37534  pm13.196a  44410  ichn  47381