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Mirrors > Home > MPE Home > Th. List > sbnOLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbn 2287 as of 8-Jul-2023. Negation inside and outside of substitution are equivalent. For a version requiring disjoint variables, but fewer axioms, see sbnvOLD 2322. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbnOLD | ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsb1 2510 | . . 3 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑))) | |
2 | exanali 1859 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
3 | 2 | anbi2i 624 | . . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
4 | annim 406 | . . 3 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
5 | 1, 3, 4 | 3bitri 299 | . 2 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
6 | dfsb3 2533 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
7 | 5, 6 | xchbinxr 337 | 1 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃wex 1780 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: (None) |
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