Theorem sbnOLD 2495

Description: Obsolete version of sbn 2253 as of 8-Jul-2023. Negation inside and outside of substitution are equivalent. For a version requiring disjoint variables, but fewer axioms, see sbnvOLD 2287. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sbnOLD

Step	Hyp	Ref	Expression
1		dfsb1 2464	. . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)))
2		exanali 1840	. . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	2	anbi2i 622	. . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)))
4		annim 404	. . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	1, 3, 4	3bitri 298	. 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6		dfsb3 2487	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	5, 6	xchbinxr 336	1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520  ∃wex 1761  [wsb 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043

This theorem is referenced by: (None)