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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. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) Revise df-sb 2071. (Revised by BJ, 25-Dec-2020.) |
Ref | Expression |
---|---|
sbn | ⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 2071 | . 2 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))) | |
2 | alinexa 1848 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
3 | 2 | imbi2i 335 | . . 3 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ (𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
4 | 3 | albii 1825 | . 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
5 | alinexa 1848 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ¬ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
6 | dfsb7 2279 | . . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
7 | 5, 6 | xchbinxr 334 | . 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ¬ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ¬ [𝑡 / 𝑥]𝜑) |
8 | 1, 4, 7 | 3bitri 296 | 1 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1539 ∃wex 1785 [wsb 2070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-10 2140 ax-12 2174 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ex 1786 df-nf 1790 df-sb 2071 |
This theorem is referenced by: sbex 2281 sbi2 2302 sbor 2307 sbcng 3769 difab 4239 wl-sb8et 35687 pm13.196a 41985 ichn 44860 |
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