Theorem sbnvOLD 2287

Description: Obsolete version of sbn 2253 as of 8-Jul-2023. Substitution is not affected by negation. Version of sbn 2253 with a disjoint variable condition, not requiring ax-13 2344. (Contributed by Wolf Lammen, 18-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbnvOLD

Step	Hyp	Ref	Expression
1		exanali 1840	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb5 2240	. 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑))
3		sb6 2066	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
4	3	notbii 321	. 2 ⊢ (¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	1, 2, 4	3bitr4i 304	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

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:  sbi2vOLD  2289