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Mirrors > Home > MPE Home > Th. List > sb4b | Structured version Visualization version GIF version |
Description: Simplified definition of substitution when variables are distinct. Version of sb6 2089 with a distinctor. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 27-May-1997.) Revise df-sb 2066. (Revised by Wolf Lammen, 21-Feb-2024.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb4b | ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfna1 2152 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑡 | |
2 | nfeqf2 2391 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡) | |
3 | 1, 2 | nfan1 2195 | . . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) |
4 | equequ2 2029 | . . . . . . 7 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡)) | |
5 | 4 | imbi1d 344 | . . . . . 6 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑))) |
6 | 5 | adantl 484 | . . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑))) |
7 | 3, 6 | albid 2219 | . . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
8 | 7 | pm5.74da 802 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
9 | 8 | albidv 1917 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
10 | df-sb 2066 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
11 | ax6ev 1968 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑡 | |
12 | 11 | a1bi 365 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
13 | 19.23v 1939 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) | |
14 | 12, 13 | bitr4i 280 | . 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
15 | 9, 10, 14 | 3bitr4g 316 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 ∃wex 1776 [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-12 2172 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 |
This theorem is referenced by: sb3b 2497 sb2 2500 sb4OLD 2502 sb1OLD 2503 sb4a 2505 hbsb2 2517 dfsb2 2528 sbcom3 2544 sbal1 2568 sbal2 2569 sbal2OLD 2570 wl-2sb6d 34788 wl-sbalnae 34792 |
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