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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 2091 with a distinctor. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 27-May-1997.) Revise df-sb 2071. (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 2153 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑡 | |
2 | nfeqf2 2376 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡) | |
3 | 1, 2 | nfan1 2198 | . . . . 5 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) |
4 | equequ2 2034 | . . . . . . 7 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡)) | |
5 | 4 | imbi1d 345 | . . . . . 6 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑))) |
6 | 5 | adantl 485 | . . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑))) |
7 | 3, 6 | albid 2220 | . . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑡 ∧ 𝑦 = 𝑡) → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
8 | 7 | pm5.74da 804 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
9 | 8 | albidv 1928 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) |
10 | df-sb 2071 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
11 | ax6ev 1978 | . . . 4 ⊢ ∃𝑦 𝑦 = 𝑡 | |
12 | 11 | a1bi 366 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
13 | 19.23v 1950 | . . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) | |
14 | 12, 13 | bitr4i 281 | . 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
15 | 9, 10, 14 | 3bitr4g 317 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 ∃wex 1787 [wsb 2070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-12 2175 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-sb 2071 |
This theorem is referenced by: sb3b 2476 sb2 2479 sb1OLD 2481 sb4a 2483 hbsb2 2485 dfsb2 2496 sbcom3 2509 sbal1 2532 sbal2 2533 wl-2sb6d 35450 wl-sbalnae 35454 |
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