Theorem sbn 2522

Description: Negation inside and outside of substitution are equivalent. For a version requiring disjoint variables, but fewer axioms, see sbnv 2334. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)

Assertion

Ref	Expression
sbn	⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbn

Step	Hyp	Ref	Expression
1		df-sb 2068	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)))
2		exanali 1959	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	2	anbi2i 616	. . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		annim 394	. . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	1, 3, 4	3bitri 289	. 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6		dfsb3 2505	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	5, 6	xchbinxr 327	1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1654  ∃wex 1878  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-12 2220  ax-13 2389

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-sb 2068

This theorem is referenced by:  sbi2  2524  sbor  2529  sban  2530  sbex  2597  sbcng  3703  difab  4127  bj-ab0  33422  wl-sb8et  33878  pm13.196a  39453