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Mirrors > Home > ILE Home > Th. List > sbal1 | GIF version |
Description: A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor ¬ ∀xx = z. (Contributed by NM, 5-Aug-1993.) (Proof rewitten by Jim Kingdon, 24-Feb-2018.) |
Ref | Expression |
---|---|
sbal1 | ⊢ (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbal 1873 | . . . 4 ⊢ ([w / y]∀xφ ↔ ∀x[w / y]φ) | |
2 | 1 | sbbii 1645 | . . 3 ⊢ ([z / w][w / y]∀xφ ↔ [z / w]∀x[w / y]φ) |
3 | sbal1yz 1874 | . . 3 ⊢ (¬ ∀x x = z → ([z / w]∀x[w / y]φ ↔ ∀x[z / w][w / y]φ)) | |
4 | 2, 3 | syl5bb 181 | . 2 ⊢ (¬ ∀x x = z → ([z / w][w / y]∀xφ ↔ ∀x[z / w][w / y]φ)) |
5 | ax-17 1416 | . . 3 ⊢ (∀xφ → ∀w∀xφ) | |
6 | 5 | sbco2v 1818 | . 2 ⊢ ([z / w][w / y]∀xφ ↔ [z / y]∀xφ) |
7 | ax-17 1416 | . . . 4 ⊢ (φ → ∀wφ) | |
8 | 7 | sbco2v 1818 | . . 3 ⊢ ([z / w][w / y]φ ↔ [z / y]φ) |
9 | 8 | albii 1356 | . 2 ⊢ (∀x[z / w][w / y]φ ↔ ∀x[z / y]φ) |
10 | 4, 6, 9 | 3bitr3g 211 | 1 ⊢ (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∀wal 1240 [wsb 1642 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 |
This theorem is referenced by: (None) |
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