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Mirrors > Home > MPE Home > Th. List > nfsb4t | Structured version Visualization version GIF version |
Description: A variable not free in a proposition remains so after substitution in that proposition with a distinct variable (closed form of nfsb4 2503). Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfsb4t | ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12 2249 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
2 | 1 | sps 2183 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
3 | 2 | drnf2 2447 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
4 | 3 | biimpd 229 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
5 | 4 | spsd 2185 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
6 | 5 | impcom 407 | . . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
7 | 6 | a1d 25 | . 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
8 | nfnf1 2152 | . . . . 5 ⊢ Ⅎ𝑧Ⅎ𝑧𝜑 | |
9 | 8 | nfal 2322 | . . . 4 ⊢ Ⅎ𝑧∀𝑥Ⅎ𝑧𝜑 |
10 | nfnae 2437 | . . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦 | |
11 | 9, 10 | nfan 1897 | . . 3 ⊢ Ⅎ𝑧(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
12 | nfa1 2149 | . . . 4 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧𝜑 | |
13 | nfnae 2437 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
14 | 12, 13 | nfan 1897 | . . 3 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
15 | sp 2181 | . . . 4 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜑) | |
16 | 15 | adantr 480 | . . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜑) |
17 | nfsb2 2486 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | |
18 | 17 | adantl 481 | . . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥[𝑦 / 𝑥]𝜑) |
19 | 1 | a1i 11 | . . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))) |
20 | 11, 14, 16, 18, 19 | dvelimdf 2452 | . 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
21 | 7, 20 | pm2.61dan 813 | 1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 Ⅎwnf 1780 [wsb 2062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 |
This theorem is referenced by: nfsb4 2503 nfsbd 2525 |
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