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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 2536). Usage of this theorem is discouraged because it depends on ax-13 2386. (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 2248 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
2 | 1 | sps 2179 | . . . . . . 7 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
3 | 2 | drnf2 2462 | . . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
4 | 3 | biimpd 231 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
5 | 4 | spsd 2181 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
6 | 5 | impcom 410 | . . 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 2154 | . . . . 5 ⊢ Ⅎ𝑧Ⅎ𝑧𝜑 | |
9 | 8 | nfal 2338 | . . . 4 ⊢ Ⅎ𝑧∀𝑥Ⅎ𝑧𝜑 |
10 | nfnae 2452 | . . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦 | |
11 | 9, 10 | nfan 1896 | . . 3 ⊢ Ⅎ𝑧(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
12 | nfa1 2151 | . . . 4 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑧𝜑 | |
13 | nfnae 2452 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦 | |
14 | 12, 13 | nfan 1896 | . . 3 ⊢ Ⅎ𝑥(∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) |
15 | sp 2177 | . . . 4 ⊢ (∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧𝜑) | |
16 | 15 | adantr 483 | . . 3 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜑) |
17 | nfsb2 2518 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | |
18 | 17 | adantl 484 | . . 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 2467 | . 2 ⊢ ((∀𝑥Ⅎ𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
21 | 7, 20 | pm2.61dan 811 | 1 ⊢ (∀𝑥Ⅎ𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 Ⅎwnf 1780 [wsb 2065 |
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 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2156 ax-12 2172 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 |
This theorem is referenced by: nfsb4 2536 nfsbd 2560 ichnfimlem1 43614 |
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