Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sbco2 | GIF version |
Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Ref | Expression |
---|---|
sbco2.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sbco2 | ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | nfri 1412 | . 2 ⊢ (𝜑 → ∀𝑧𝜑) |
3 | 2 | sbco2h 1838 | 1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 Ⅎwnf 1349 [wsb 1645 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: nfsbt 1850 sb7af 1869 sbco4lem 1882 sbco4 1883 eqsb3 2141 clelsb3 2142 clelsb4 2143 sb8ab 2159 sbralie 2546 sbcco 2785 |
Copyright terms: Public domain | W3C validator |