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Mirrors > Home > MPE Home > Th. List > sbn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
sbn | ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) |
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) |
Colors of variables: wff setvar class |
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 |
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