Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dfsb2 | Structured version Visualization version GIF version |
Description: An alternate definition of proper substitution that, like dfsb1 2486, mixes free and bound variables to avoid distinct variable requirements. Usage of this theorem is discouraged because it depends on ax-13 2373. (Contributed by NM, 17-Feb-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb2 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2179 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝑥 = 𝑦) | |
2 | sbequ2 2244 | . . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑)) | |
3 | 2 | sps 2181 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑)) |
4 | orc 863 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
5 | 1, 3, 4 | syl6an 680 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
6 | sb4b 2476 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
7 | olc 864 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
8 | 6, 7 | syl6bi 252 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
9 | 5, 8 | pm2.61i 182 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
10 | sbequ1 2243 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
11 | 10 | imp 406 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑) |
12 | sb2 2481 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | |
13 | 11, 12 | jaoi 853 | . 2 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)) → [𝑦 / 𝑥]𝜑) |
14 | 9, 13 | impbii 208 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∀wal 1539 [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 ax-13 2373 |
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: dfsb3 2499 |
Copyright terms: Public domain | W3C validator |